Datasets:

AnkitSatpute

/

zbMath_allft

like 0

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces degree of the fibre; rational fibration; ruled surfaces Rational and ruled surfaces, Families, fibrations in algebraic geometry On complex ruled surfaces. I

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces canonical hyperplane sections; classification of surfaces; only one effective anticanonical divisor on minimal resolution of singularities; elliptic ruled surfaces Epema, D.: Surfaces with canonical hyperplane sections, Thèse Leiden, cf. aussi: Indagationes Math.45, 173-184 (1983) Families, moduli, classification: algebraic theory, Divisors, linear systems, invertible sheaves, Global theory and resolution of singularities (algebro-geometric aspects), Special surfaces, Singularities of surfaces or higher-dimensional varieties Surfaces with canonical hyperplane sections

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces places of algebraic function fields; description of holomorphy ring of function fields; proof of Ax-Kochen-Ershov theorem; approximation theorems Kuhlmann, F. -V.; Prestel, A.: On places of algebraic function fields. J. reine angew. Math. 353, 181-195 (1984) General valuation theory for fields, Arithmetic theory of algebraic function fields, Model theory of fields, Transcendental field extensions, Real algebraic and real-analytic geometry, Model-theoretic algebra, Local ground fields in algebraic geometry Places of algebraic function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Drinfeld quasi-modular forms; Hankel determinants; function fields of positive characteristic Modular forms associated to Drinfel'd modules, Global ground fields in algebraic geometry, Formal groups, \(p\)-divisible groups Hankel-type determinants and Drinfeld quasi-modular forms

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Mickelsson-Faddeev cocycle; existence of string structures; bundle gerbe; quantum field theory; Atiyah-Patodi-Singer index theory; bundle of fermionic Fock spaces; gauge group action; Dixmier-Douady class; fermions in external fields; APS theorem; WZW model; Riemann surfaces; global Hamiltonian anomalies A. Alan Carey, A. Mickelsson, and M. Murray, ''Bundle gerbes applied to quantum field theory,'' hep-th/9711133. Quantum field theory; related classical field theories, Applications of global differential geometry to the sciences, Moduli problems for topological structures, Applications of PDEs on manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Generalizations of fiber spaces and bundles in algebraic topology Bundle gerbes applied to quantum field theory

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces explicit formulae of prime number theory; Riemann zeta-function; Poisson summation formula; Riemann hypothesis; Hadamard product formula; zeros; prime number theorem; Lindelöf hypothesis; zeta-functions attached to curves over finite fields; approximate functional equation; large number of exercises Patterson, S.J. (1988). An Introduction to the Theory of the Riemann Zeta-Function. Cambridge Studies in Advanced Mathematics 14 . Cambridge: Cambridge Univ. Press. \(\zeta (s)\) and \(L(s, \chi)\), Research exposition (monographs, survey articles) pertaining to number theory, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry An introduction to the theory of the Riemann zeta-function

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces products of conjugates of generators; algorithms; Garside normal forms; quasipositivity in braid groups; plane real algebraic curves; trigonal real pseudoholomorphic curves; rational ruled surfaces S Y Orevkov, Quasipositivity problem for \(3\)-braids, Turkish J. Math. 28 (2004) 89 Braid groups; Artin groups, Word problems, other decision problems, connections with logic and automata (group-theoretic aspects), Generators, relations, and presentations of groups, Plane and space curves, Symbolic computation and algebraic computation Quasipositivity problem for 3-braids.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces twistor spaces; slice regular functions; functions of hypercomplex variables; rational and ruled surfaces; slice-polynomial functions Twistor methods in differential geometry, Functions of hypercomplex variables and generalized variables, Other generalizations of function theory of one complex variable, Rational and ruled surfaces Slice-polynomial functions and twistor geometry of ruled surfaces in \(\mathbb {CP}^3\)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces diophantine equations; Siegel's theorem; integral points on affine curves; function-fields of characteristic zero José Felipe Voloch, Siegel's theorem for complex function fields, Proc. Amer. Math. Soc. 121 (1994), no. 4, 1307 -- 1308. Elliptic curves over global fields, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Global ground fields in algebraic geometry Siegel's theorem for complex function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Weierstrass function; tau function; Jacobians; finite-gap potentials; Korteweg-de Vries equation; Kadomtsev-Petviashvili equation; KP equation; KdV equation; integrable differential equations; theta functions of Riemann surfaces; elliptic finite-gap solutions; soliton equations --, New elliptic potentials.Acta Appl. Math., 36 (1994), 27--48. Special algebraic curves and curves of low genus, KdV equations (Korteweg-de Vries equations), Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Elliptic curves, Theta functions and abelian varieties New elliptic potentials

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces abstract elliptic function fields; divisor class group of finite order; automorphisms; meromorphisms; addition theorems; structure of ring of meromorphisms; Riemann hypothesis Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Zur Theorie der abstrakten elliptischen Funktionenkörper. I: Die Struktur der Gruppe der Divisorenklassen endlicher Ordnung. II: Automorphismen und Meromorphismen. Das Additionsproblem. III: Die Struktur des Meromorphismenrings. Die Riemannsche Ver\-mutung.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces cohomology of finite Chevalley groups; cohomology stability; connected split reductive group scheme; change of fields; algebra retract; elementary abelian \(\ell \)-subgroups; cohomology algebras; integral cohomology; cohomological restriction map Friedlander, E.: Multiplicative stability for the cohomology of finite Chevalley groups. Comment. Math. Helv.63, 108--113 (1988). Erratum: Comment. Math. Helv.64, 348 (1989) Cohomology theory for linear algebraic groups, Linear algebraic groups over finite fields, Homology of classifying spaces and characteristic classes in algebraic topology, Group schemes, Cohomology of groups Multiplicative stability for the cohomology of finite Chevalley groups

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Teichmüller modular function fields; pro-\(l\) number field towers; moduli stack of smooth projective curves; stability; braid groups Nakamura, H.; Takao, N.; Ueno, R., Some stability properties of Teichmüller modular function fields with pro-\textit{} weight structures, Math. ann., 302, 197-213, (1995), MR 96h:14041 Arithmetic ground fields for curves, Coverings of curves, fundamental group, Families, moduli of curves (algebraic), Braid groups; Artin groups Some stability properties of Teichmüller modular function fields with pro-\(l\) weight structures

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic cycles; hyper-Kähler varieties; moduli spaces of sheaves on \(K3\) surfaces; birational motives; abelian varieties; co-algebras (Equivariant) Chow groups and rings; motives, Algebraic cycles, Holomorphic symplectic varieties, hyper-Kähler varieties, Abelian varieties and schemes, Rational and birational maps On the birational motive of hyper-Kähler varieties

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces division algebra; value group; center; Brauer group; tensor products of symbol algebras; Dubrovin valuation rings; armatures; central simple algebras Adrian R. Wadsworth, Valuations on tensor products of symbol algebras, Azumaya algebras, actions, and modules (Bloomington, IN, 1990) Contemp. Math., vol. 124, Amer. Math. Soc., Providence, RI, 1992, pp. 275 -- 289. Valuations, completions, formal power series and related constructions (associative rings and algebras), Finite-dimensional division rings, Brauer groups of schemes, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Valuations on tensor products of symbol algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces towers of function fields; rational points; finite fields; hypergeometric functions; Deuring's polynomial Hasegawa, On asymptotically optimal towers over quadratic fields related to Gauss hypergeometric functions, Int. J. Number Theory 6 pp 989-- (2010) Arithmetic theory of algebraic function fields, Curves over finite and local fields, Algebraic coding theory; cryptography (number-theoretic aspects), Finite ground fields in algebraic geometry, Applications to coding theory and cryptography of arithmetic geometry, Classical hypergeometric functions, \({}_2F_1\) On asymptotically optimal towers over quadratic fields related to Gauss hypergeometric functions

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces generic matrices; stable rationality; fields of invariants; group algebras; lattices Beneish, E.: Induction Theorems on the center of the ring of generic matrices. Trans. Am. Math. Soc. 350(9), 3571--3585 (1998) Semiprime p.i. rings, rings embeddable in matrices over commutative rings, Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Integral representations of finite groups, Group actions on varieties or schemes (quotients) Induction theorems on the stable rationality of the center of the ring of generic matrices

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces function fields; irreducible polynomials; hyperelliptic curves; derivatives of \(L\)-functions; moments of \(L\)-functions; quadratic Dirichlet \(L\)-functions; random matrix theory Zeta and \(L\)-functions in characteristic \(p\), \(\zeta (s)\) and \(L(s, \chi)\), Curves over finite and local fields, Relations with random matrices, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Mean values of derivatives of \(L\)-functions in function fields. III

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces linear systems of divisor on \(r\)-fold symmetric products of elliptic curves; surfaces of general type; moduli of surfaces; Kodaira-Spencer map; Kuranishi family Catanese, F.; Ciliberto, C., Symmetric products of elliptic curves and surfaces of general type with \(p_g=q=1\), J. Algebr. Geom., 2, 389-411, (1993) Families, moduli, classification: algebraic theory, Divisors, linear systems, invertible sheaves, Surfaces of general type, Elliptic curves Symmetric products of elliptic curves and surfaces of general type with \(p_ g=q=1\)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Waring rank; symmetric tensors; homogeneous polynomials; ideals of points; Hilbert function; apolarity Polynomials, factorization in commutative rings, Secant varieties, tensor rank, varieties of sums of powers, Polynomial rings and ideals; rings of integer-valued polynomials, Projective techniques in algebraic geometry On minimal decompositions of low rank symmetric tensors

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces elliptic curves over function fields; explicit computation of \(L\)-functions; special values of \(L\)-functions and BSD conjecture; estimates of special values; analogue of the Brauer-Siegel theorem Elliptic curves over global fields, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Zeta and \(L\)-functions in characteristic \(p\) Explicit \(L\)-functions and a Brauer-Siegel theorem for Hessian elliptic curves

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces deformation types of arbitrary ruled surfaces; no moduli scheme parametrizing all ruled surfaces with given Chern numbers 10.1515/crll.1992.426.203 Rational and ruled surfaces, Formal methods and deformations in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Deformations of ruled surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces function fields; Bombieri-lang conjecture; varieties of general type Rational points, Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry Rational points of varieties with ample cotangent bundle over function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces quadratic forms; \(u\)-invariant; power series fields; function fields of curves; orderings of fields; patching of fields Scheiderer, Claus: The u-invariant of one-dimensional function fields over real power series fields, Arch. math. (Basel) 93, No. 3, 245-251 (2009) Algebraic theory of quadratic forms; Witt groups and rings, Quadratic forms over general fields, Algebraic functions and function fields in algebraic geometry The \(u\)-invariant of one-dimensional function fields over real power series fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces class numbers; function fields; mean values of \(L\)-functions Andrade, J. C., A note on the mean value of \textit{L}-functions in function fields, Int. J. Number Theory, 8, 7, 1725-1740, (2012) Class numbers, class groups, discriminants, Arithmetic theory of algebraic function fields, Curves over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) A note on the mean value of \(L\)-functions in function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic function fields; finite field of constants; Severi's algebraic theory of correspondences; Hurwitz's transcendental theory; group of divisor classes; Riemann hypothesis for function fields; action of Galois group André Weil, Sur les fonctions algébriques à corps de constantes fini, C. R. Acad. Sci. Paris 210 (1940), 592 -- 594 (French). Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Arithmetic ground fields for curves, Finite ground fields in algebraic geometry Sur les fonctions algébriques à corps de constantes fini

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces noncommutative tori; real multiplication; Stark numbers; real quadratic fields; spectral triples; noncommutative boundary of modular curves; modular shadows; quantum statistical mechanics Noncommutative topology, Relations with noncommutative geometry, Noncommutative geometry (à la Connes), Quantum dynamics and nonequilibrium statistical mechanics (general), Noncommutative algebraic geometry Noncommutative geometry and arithmetic

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces \(p\)-extensions of algebraic function fields; Artin-Schreier theory; characteristic \(p\); genus; number of rational points; coding theory; gap number Arnaldo Garcia and Henning Stichtenoth, Elementary abelian \(p\)-extensions of algebraic function fields, Manuscr. Math. 72 (1991), 67--79. Algebraic functions and function fields in algebraic geometry, Arithmetic theory of algebraic function fields, Geometric methods (including applications of algebraic geometry) applied to coding theory, Riemann surfaces; Weierstrass points; gap sequences, Computational aspects of algebraic curves Elementary abelian \(p\)-extensions of algebraic function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces \(K_ 2\) of fields; Brauer group; cyclic algebras; generators A. S. Merkurjev, Structure of the Brauer group of fields , Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), no. 4, 828-846, 895, trad. anglaise, Math. USSR-Izv. 27 (1986), 141-157. Skew fields, division rings, Brauer groups of schemes, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Galois cohomology On the structure of the Brauer group of a field

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces noncommutative projective geometry; noncommutative surfaces; Sklyanin algebras; noncommutative blowing up; Noetherian graded rings; sporadic ideals; divisor layering; graded quotient ring; twisted homogeneous coordinate ring; elliptic algebra; exceptional line modules; Godie torsion module D. Rogalski, S. J. Sierra and J. T. Stafford, Noncommutative blowups of elliptic algebras, Algebr. Represent. Theory, (2014), 1--39.Zbl 06445654 MR 3336351 Noncommutative algebraic geometry, Elliptic curves, Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.), Noetherian rings and modules (associative rings and algebras), Rings arising from noncommutative algebraic geometry, Graded rings and modules (associative rings and algebras) Noncommutative blowups of elliptic algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Automorphism of function fields; singular points; rational function fields. Automorphisms of curves, Arithmetic theory of algebraic function fields, Separable extensions, Galois theory, Algebraic functions and function fields in algebraic geometry A relation between Galois automorphism and curve singularity

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces rank of abelian variety; function fields; elliptic curve Pacheco, A.: The rank of abelian varieties over function fields. Manuscripta Math. 118, 361--381 (2005) Abelian varieties of dimension \(> 1\), Rational points, Arithmetic ground fields for curves, Arithmetic ground fields for abelian varieties On the rank of abelian varieties over function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces fields of meromorphic function; Mittag-Leffler theorem; Weierstrass theorem P. Cutillas Ripoll,Construction of certain function fields associated with a compact Riemann surface, American Journal of Mathematics106 (1984), 1423--1450. Compact Riemann surfaces and uniformization, Algebraic functions and function fields in algebraic geometry Construction of certain function fields associated with a compact Riemann surface

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces skew-symmetric function; determinant; decomposable; Plücker relation Duzhin, S, No article title, Decomposable skew-symmetric functions. Moscow Math. J., 3, 881-888, (2003) Symmetric functions and generalizations, Determinants, permanents, traces, other special matrix functions, Analytical algebras and rings, Grassmannians, Schubert varieties, flag manifolds Decomposable skew-symmetric functions

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Hall algebras; elliptic curves; finite fields; derived categories; flat deformations; symmetric polynomials; coherent sheaves; Drinfeld doubles I. Burban and O. Schiffmann, On the Hall algebra of an elliptic curve. I, \textit{Duke Math. J.}, 161(2012), no. 7, 1171--1231.Zbl 1286.16029 MR 2922373 Hopf algebras and their applications, Elliptic curves, Quantum groups (quantized enveloping algebras) and related deformations, Infinite-dimensional Lie groups and their Lie algebras: general properties, Representations of quivers and partially ordered sets, Ring-theoretic aspects of quantum groups On the Hall algebra of an elliptic curve. I.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces anabelian geometry; pro-\(\ell \) groups; Galois theory; function fields; valuations theory; (Riemann) space of prime divisors; Hilbert decomposition theory; Parshin chains; decomposition graphs Pop, F., Recovering function fields from their decomposition graphs, 519-594, (2012), New York Field arithmetic, Separable extensions, Galois theory, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Algebraic functions and function fields in algebraic geometry Recovering function fields from their decomposition graphs

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic curves; algebraic function fields; maximal curves; maximal function fields; automorphisms of function fields Güneri, C.; Özdemir, M.; Stichtenoth, H., The automorphism group of the generalized giulietti-korchmáros function field, \textit{Adv. Geom.}, 13, 369-380, (2013) Curves over finite and local fields, Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry The automorphism group of the generalized Giulietti-Korchmáros function field

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces arithmetical ruled surfaces; curves of genus 0 Arithmetic ground fields for curves, Special surfaces, Quaternion and other division algebras: arithmetic, zeta functions, Global ground fields in algebraic geometry On arithmetical ruled surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces textbook; partially ordered sets; Zorn's lemma; number theory; fields; rings; abelian groups; polynomials; field extension; formal power series; polynomial rings; finite fields; power series; rational function; Bernoulli numbers; Puiseux series; Laurent series; ideals; quotient rings; factorization; Noetherian rings; prime ideals; principal ideal domains; cyclic groups; homomorphism; group action; quotient group; symmetric group; semidirect product; Sylow group; modules; free modules; commutative ring; Smith normal form; elementary divisor; Jordan form; Hermitian space; projective space; bilinear form; symplectic space; quadratic form; Kähler triples; quaternions; spinors Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory Algebra I. Textbook for students of mathematics. Translated from the Russian

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic groups; irreducible representations; Hopf algebras; Lie algebras; unipotent algebraic groups; tensor product; semisimple group representations; varieties; dimension theory of local rings; tangent spaces; Borel subgroups; Galois cohomology; automorphism groups; weights; universal enveloping algebra Hochschild, G.P.: Basic theory of algebraic groups and Lie algebras. Graduate Texts Math. \textbf{75}, (1981) Linear algebraic groups over arbitrary fields, Research exposition (monographs, survey articles) pertaining to group theory, Lie algebras of linear algebraic groups, Group varieties, Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Universal enveloping (super)algebras, Classical groups (algebro-geometric aspects) Basic theory of algebraic groups and Lie algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces symmetric algebra; codimension; defect; minimal numbers of generators; Krull dimension; linear representations; algebras of invariants; symmetric and alternating groups; irreducible representations Representations of finite symmetric groups, Group actions on varieties or schemes (quotients), Vector and tensor algebra, theory of invariants Invariants of symmetric and alternating groups

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Mordell-Weil group; multidimensional function fields; Néron-Tate height; Mordell-Weil rank; Jacobian; independence of some rational points T. Shioda, Constructing curves with high rank via symmetry, Amer. J. Math., to appear. Algebraic functions and function fields in algebraic geometry, Rational points, Curves of arbitrary genus or genus \(\ne 1\) over global fields Constructing curves with high rank via symmetry

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Witt group of skew-symmetric non-singular bilinear forms; real algebraic variety of dimension 4; ring of \({\mathbb{C}}\)-valued regular functions; Chow ring; algebraic hypersurfaces Witt groups of rings, Real algebraic sets, \(4\)-folds Symplectic complex bundles over real algebraic four-folds

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces twisted presheaf; quasi-compact semiseparated twisted presheaf; qcss twisted presheaf; noncommutative deformation; presheaf of noncommutative algebras; deformation of abelian categories; Hochschild cohomology; Toda's construction; Gestenhaber-schack complex; deformation quantization; deformation of algebras Dinh Van, H.; Liu, L.; Lowen, W., Non-commutative deformations and quasi-coherent modules, Selecta Math., 23, 2, 1061-1119, (2016) Deformations and infinitesimal methods in commutative ring theory, Noncommutative algebraic geometry Non-commutative deformations and quasi-coherent modules

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces X-rank; secant varieties; rank of tensor; Terracini's question; partially symmetric tensors; Comon's conjecture Buczyński, J.; Landsberg, J.M.; Ranks of tensors and a generalization of secant varieties; Linear Algebra Appl.: 2013; Volume 438 ,668-689. Multilinear algebra, tensor calculus, Effectivity, complexity and computational aspects of algebraic geometry, Canonical forms, reductions, classification, Vector spaces, linear dependence, rank, lineability Ranks of tensors and a generalization of secant varieties

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces free energy; Euler characteristic of the moduli space; punctured compact Riemann surfaces; Penner's connected generating function N. Chair, Rev. Math. Phys. 3 pp 285-- (1991) Riemann surfaces, Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic) The (orbifold) Euler characteristic of the moduli space of curves and the continuum limit of Penner's connected generating function

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces effective geometric Bogomolov conjecture; curves of genus 3 over function fields; self-intersection of the relative dualizing sheaf; admissible constants of the metrized dual graph K. Yamaki, Geometric Bogomolov's conjecture for curves of genus 3 over function fields, J. Math. Kyoto Univ. 42 (2002), 57-81. Varieties over global fields, Global ground fields in algebraic geometry Geometric Bogomolov's conjecture for curves of genus 3 over function fields.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Waring rank; tensor rank; symmetric tensor; sum of powers of linear forms; ternary form De Paris, A.; High-rank ternary forms of even degree; Arch. Math.: 2017; Volume 109 ,505-510. Canonical forms, reductions, classification, Classical problems, Schubert calculus, Computational aspects and applications of commutative rings, Multilinear algebra, tensor calculus, Quadratic and bilinear forms, inner products High-rank ternary forms of even degree

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces rational points; inseparable extensions of function field; Mordell conjecture for number fields; genus drop; prime characteristic; non-conservative curves Voloch, J. F.: A Diophantine problem on algebraic curves over function fields of positive characteristic. Bull. soc. Math. France 119, 121-126 (1991) Rational points, Finite ground fields in algebraic geometry, Arithmetic theory of algebraic function fields, Global ground fields in algebraic geometry, Curves in algebraic geometry A Diophantine problem on algebraic curves over function fields of positive characteristic

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces towers of function fields; asymptotically good towers; Drinfeld-Vladut bound; Artin-Schreier extension; long algebraic-geometric codes with good parameters García, A.; Stichtenoth, H., On the asymptotic behaviour of some towers of function fields over finite fields, J. Number Theory, 61, 248-273, (1996) Arithmetic theory of algebraic function fields, Curves over finite and local fields, Finite ground fields in algebraic geometry, Arithmetic ground fields for curves, Geometric methods (including applications of algebraic geometry) applied to coding theory On the asymptotic behaviour of some towers of function fields over finite fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces geometric Langlands program; Langlands correspondence for function fields; moduli stack of \(G\)-bundles; Drinfeld-Lafforgue-Vinberg compactification; singularities of the degeneration; miraculous duality of Drinfeld and Gaitsgory; Drinfeld-Wang strange invariant bilinear form S. Schieder, Picard-Lefschetz oscillators for the Drinfeld-Lafforgue-Vinberg degeneration for \(\text{SL}_{2}\), Duke Math. J. 167 (2018), 835--921. Geometric Langlands program (algebro-geometric aspects) Picard-Lefschetz oscillators for the Drinfeld-Lafforgue-Vinberg degeneration for \(\mathrm{SL}_2\)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces ruled surfaces; fat points; Hilbert function DOI: 10.2478/BF02475962 Divisors, linear systems, invertible sheaves, Rational and ruled surfaces, Projective techniques in algebraic geometry Zero-dimensional subschemes of ruled varieties

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Proceedings; Symposium; Kyoto (Japan); Frobenius mappings; Commutative rings; rings with approximation property; unramified coverings; coverings of algebraic surfaces; CM modules; symbolic Rees algebras; ASL domains Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Proceedings, conferences, collections, etc. pertaining to commutative algebra, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings of conferences of miscellaneous specific interest, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Cohen-Macaulay modules, Minimal model program (Mori theory, extremal rays), Rings with straightening laws, Hodge algebras Applications of Frobenius mappings to the theory of commutative rings. Proceedings of a symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, September 18-22, 1989

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic surfaces; action of finite groups; resolution of singularities; Heisenberg algebra; Virasoro algebra; representation of Lie algebras; vertex algebras; equivariant \(K\)-theory of schemes W. Wang, Algebraic structures behind Hilbert schemes and wreath products, in: S. Berman et al. (Eds.), Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory, Charlottesville, VA, 2000; Contemp. Math. 297 (2002) 271-295. Parametrization (Chow and Hilbert schemes), Virasoro and related algebras, Vertex operators; vertex operator algebras and related structures, Extensions, wreath products, and other compositions of groups Algebraic structures behind Hilbert schemes and wreath products.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces non-archimedean valued fields; analytic functions; \(p\)-adic cohomology; Weil conjectures; \(p\)-adic analytic varieties; action of Frobenius; rigid cohomology; \(p\)-adic analytic functions; Morita's \(p\)-adic gamma function \(p\)-adic differential equations, Rigid analytic geometry, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Non-Archimedean function theory \(p\)-adic differential equations and \(p\)-adic interpolation of classical formulae

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Krichever-Novikov algebras; meromorphic vector fields; theta functions; Weierstraß \(\sigma \) function M. Schlichenmaier, ''Krichever-Novikov Algebras for More Than Two Points: Explicit Generators,'' Lett. Math. Phys. 19, 327--336 (1990). Differentials on Riemann surfaces, Theta functions and curves; Schottky problem, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Krichever-Novikov algebras for more than two points: explicit generators

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces p-adic L-functions; CM fields; totally complex quadratic extension of a totally real field; Grössencharacter; p-adic measure; p-adic interpolation of Hecke L-function; functional equation; non-analytic Eisenstein series; Hilbert modular group; p-adic differential operators; p-adic Eisenstein series N.M. Katz, ''p-Adic L-functions for CM-fields,'' Invent. Math. 49(3), 199--297 (1978). Zeta functions and \(L\)-functions, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), \(p\)-adic differential equations, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Complex multiplication and moduli of abelian varieties \(p\)-adic \(L\)-functions for CM fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Koszul complex; homomorphism of symmetric algebras; almost complete intersection; Rees ring Restuccia G.,Formes linéaires et algèbres symétriques, Bull. Sc. Math.,110 (1986), 391--410. Homological conjectures (intersection theorems) in commutative ring theory, Complete intersections, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Linear forms and symmetric algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Clifford algebras; Euler characteristics; Quadratic fibrations; mirror symmetry; string theory; Calabi-Yau manifolds; nonlinear sigma models; Calabi-Yau manifolds; superstring theory; mirror symmetric pair; symplectic; Fukaya category; bounded derived category; Homological Projective Duality; complete intersection of quadrics; Lefshetz decomposition; non-commutative algebraic variety; Clifford non-commutative varieties; Hodge numbers; Batyrev's stringy Hodge numbers; Clifford-stringy Euler characteristics Noncommutative algebraic geometry, Relationships between algebraic curves and physics, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies On stringy Euler characteristics of Clifford non-commutative varieties

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Sklyanin algebras; Grothendieck categories; noncommutative curves; noncommutative projective geometry; graded rings; full subcategories; categories of graded modules; Krull dimension; non-commutative schemes; quasi-schemes; quasi-coherent sheaves Rings arising from noncommutative algebraic geometry, Module categories in associative algebras, Noncommutative algebraic geometry, Homological dimension in associative algebras, Graded rings and modules (associative rings and algebras) Curves in Grothendieck categories.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic cycles over finite fields; zeta function; Riemann-Roch; zeta functions of zero cycles Wan, D.: Zeta functions of algebraic cycles over finite fields. Manuscripta math. 74, 413-444 (1992) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Algebraic cycles, Finite ground fields in algebraic geometry, Varieties over finite and local fields, Other Dirichlet series and zeta functions Zeta functions of algebraic cycles over finite fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces products of CM elliptic curves; Coleman's conjecture; endomorphism algebras; singular abelian surfaces Arithmetic aspects of modular and Shimura varieties, Complex multiplication and moduli of abelian varieties, Complex multiplication and abelian varieties Endomorphism algebras of geometrically split abelian surfaces over \(\mathbb{Q} \)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces sums of squares; positive semidefinite function; formally real field; real holomorphy ring; Hilbert's 17-th problem; rational function fields; real closed field Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Ordered fields, Valued fields, Valuations and their generalizations for commutative rings, Real algebraic and real-analytic geometry On a variation of Hilbert's 17th problem

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces tangent sheaves of generalized Raynaud surfaces; surface of general type; dimension of the space of vector fields; global differential 1-forms; cuspidal fibrations Takeda, Y.: Vector fields and differential forms on generalized raynaud surfaces. Tôhoku math. J. 44, 359-364 (1992) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Surfaces of general type Vector fields and differential forms on generalized Raynaud surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces noncommutative algebraic geometry; real spectrum; skew fields; matrix orderings; epic \(R\)-fields Topological and ordered rings and modules, Noncommutative algebraic geometry, Infinite-dimensional and general division rings, Ordered rings, algebras, modules, Endomorphism rings; matrix rings Orderings for noncommutative rings

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces energy function; tensor products of Kirillov-Reshetikhin crystals T. LAM AND P.PYLYAVSKYY: Intrinsic energy is a loop Schur function, preprint, 2009; arXiv:1003.3948. Graphs and linear algebra (matrices, eigenvalues, etc.), Applications of graph theory, Symmetric functions and generalizations, Combinatorial aspects of representation theory, Molecular structure (graph-theoretic methods, methods of differential topology, etc.), Tropical geometry Intrinsic energy is a loop Schur function

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces low degree polynomial equations; complexity of the equations; Nullstellensatz; birational map; rational variety; Seiberg-Witten theory; counting function; characterization of rational surfaces; minimal models J. Kollár , 'Low degree polynomial equations: arithmetic, geometry and topology', European Congress of Mathematics I (Budapest, 1996), Progress in Mathematics 168 (Birkhäuser, Basel, 1998) 255--288. Relevant commutative algebra, Polynomial rings and ideals; rings of integer-valued polynomials, Rational and ruled surfaces Low degree polynomial equations: Arithmetic, geometry and topology

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces tower of function fields; one-point code; minimum distance; Feng-Rao bound Hasegawa, T.; Kondo, S.; Kurusu, H., A sequence of one-point codes from a tower of function fields, Des. Codes Cryptogr., 41, 3, 251-267, (2006) Algebraic functions and function fields in algebraic geometry, Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory A sequence of one-point codes from a tower of function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces tensor products of symmetric powers; injective indecomposable modules; tensor products of exterior powers; tilting modules; symmetric functions; listing modules; partitions; Schur superalgebras; group schemes; representations of general linear groups; symmetric groups; Young modules Donkin, S., Symmetric and exterior powers, linear source modules and representations of Schur superalgebras, \textit{Proc. London Math. Soc.}, 83, 647-680, (2011) Representation theory for linear algebraic groups, Representations of finite symmetric groups, ``Super'' (or ``skew'') structure, Combinatorial aspects of representation theory, Group schemes Symmetric and exterior powers, linear source modules and representations of Schur superalgebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Drinfeld modules; L-series; elliptic curve; q-expansion coefficients; Poisson kernel formula; space of cusp forms; non-archimedean measure; rigid analytic space; rigid analytic modular forms for function fields; Mellin transform J. T. Teitelbaum, The Poisson kernel for Drinfeld modular curves , J. Amer. Math. Soc. 4 (1991), no. 3, 491-511. JSTOR: Drinfel'd modules; higher-dimensional motives, etc., Arithmetic theory of algebraic function fields, \(p\)-adic theory, local fields, Local ground fields in algebraic geometry, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Formal groups, \(p\)-divisible groups, Arithmetic theory of polynomial rings over finite fields The Poisson kernel for Drinfeld modular curves

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces semisimple simply connected algebraic groups; Lie algebras; fields of formal Laurent series; Iwahori subgroups; Iwahori subalgebras; finite dimensional projective varieties; dimensions; explicit formula R. Bezrukavnikov, ''The dimension of the fixed point set on affine flag manifolds,'' Math. Res. Lett., vol. 3, iss. 2, pp. 185-189, 1996. Linear algebraic groups over local fields and their integers, Grassmannians, Schubert varieties, flag manifolds, Lie algebras of linear algebraic groups The dimension of the fixed point set on affine flag manifolds

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces \(L\)-series of number fields; infinite-dimensional analogue of one- parameter group; Artin formalism; self-adjoint operator; associated heat kernel; characteristic kernel; trace; asymptotic expansion; regularized determinant of the Laplacian; Selberg zeta function; theta functions J. Jorgenson, S. Lang, Artin formalism and heat kernels. Jour. Reine. Angew. Math. 447 (1994), 165-280. Zbl0789.11055 MR1263173 Other Dirichlet series and zeta functions, Spectral theory; trace formulas (e.g., that of Selberg), Heat and other parabolic equation methods for PDEs on manifolds, Theta functions and curves; Schottky problem, Langlands \(L\)-functions; one variable Dirichlet series and functional equations, Zeta functions and \(L\)-functions of number fields, Theta functions and abelian varieties Artin formalism and heat kernels

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces elliptic curves; Mordell-Weil rank; elliptic surfaces; function fields Elliptic curves over global fields, Algebraic functions and function fields in algebraic geometry, Elliptic curves, Classical hypergeometric functions, \({}_2F_1\) The bounds of the Mordell-Weil ranks in cyclotomic towers of function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces elliptic curves over function fields; Tate-Shafarevich groups; explicit computation of \(L\)-functions; BSD conjecture; Gauss and Kloosterman sums Elliptic curves over global fields, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Gauss and Kloosterman sums; generalizations Elliptic curves with large Tate-Shafarevich groups over \(\mathbb{F}_q(t)\)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Markov equation for rational ruled surfaces; exceptional objects on rational ruled surfaces; rank of vector bundles; helix; mutation Notes on exceptional vector bundles and helices

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces noncommutative schemes; quantum planes; Artin-Schelter regular algebras; categories of graded right modules; finite-dimensional modules; Ore extensions; graded automorphisms; point modules Darin R. Stephenson, Quantum planes of weight (1,1,\?), J. Algebra 225 (2000), no. 1, 70 -- 92. Graded rings and modules (associative rings and algebras), Quantum groups (quantized enveloping algebras) and related deformations, Elliptic curves, Homological dimension in associative algebras, Ordinary and skew polynomial rings and semigroup rings, Rings arising from noncommutative algebraic geometry Quantum planes of weight \((1,1,n)\)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces skew-symmetric rank; apolarity; tensor decomposition Multilinear algebra, tensor calculus, Exterior algebra, Grassmann algebras, Factorization of matrices, Grassmannians, Schubert varieties, flag manifolds, Actions of groups and semigroups; invariant theory (associative rings and algebras), Computational aspects of associative rings (general theory) Skew-symmetric tensor decomposition

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Brauer groups; indices; function fields of projective spaces; \(P_{n,r}\)-fields Finite-dimensional division rings, Brauer groups (algebraic aspects), Algebraic functions and function fields in algebraic geometry Indices of central simple algebras over fields of functions of projective spaces.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces translations of classics (algebraic geometry); history of algebraic geometry; mathematics of the 19th century; algebraic functions; function fields; algebraic curves; Riemann-Roch theorem; algebraic differential 2.R. Dedekind, H. Weber, \(Theory of algebraic functions of one variable.\) Translated from the 1882 German original and with an introduction, bibliography and index by John Stillwell. History of Mathematics, 39. American Mathematical Society (Providence, RI; London Mathematical Society, London, 2012), pp. viii+152 History of algebraic geometry, Biographies, obituaries, personalia, bibliographies, Algebraic functions and function fields in algebraic geometry, History of mathematics in the 19th century, Arithmetic theory of algebraic function fields, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Theory of algebraic functions of one variable. Transl. from the German and introduced by John Stillwell

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces ruled surfaces; locally free sheaf of rank two Rational and ruled surfaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli On complex ruled surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces arithmetic theorem of algebraic function fields; L-function of Galois covering of curves; function-field; characteristic polynomial of the Hasse-Witt matrix; generalised Hasse-Witt invariants Cyclotomic function fields (class groups, Bernoulli objects, etc.), Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Galois theory Class groups and \(L\)-series of congruence function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces self-dual codes; algebraic geometry codes; Gilbert-Varshamov bound; Tsfasman-Vladut-Zink bound; towers of function fields; asymptotically good codes; quadratic forms; Witt's theorem Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory Self-dual codes better than the Gilbert-Varshamov bound

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces operad; bar-construction; graph; symmetric function; Feynman diagram; modular operads; moduli spaces of curves; Euler characteristics; Feynman transforms E. Getzler and M. M. Kapranov, Modular operads, \textit{Compositio Math.}, 110 (1998), no. 1, 65--126.Zbl 0894.18005 MR 1601666 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads, Families, moduli of curves (algebraic), Relational systems, laws of composition Modular operads

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces finite fields; function field; hyperelliptic curve; moments of quadratic Dirichlet L-function; prime polynomial Andrade, J. C.; Keating, J. P., Mean value theorems for \textit{L}-functions over prime polynomials for the rational function field, Acta Arith., 161, 4, 371-385, (2013) Curves over finite and local fields, Zeta and \(L\)-functions in characteristic \(p\), Relations with random matrices, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Mean value theorems for \(L\)-functions over prime polynomials for the rational function field

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces pfaffian; Azumaya algebra; quadratic form; alternating elements; discriminant module; tensor products of quaternion algebras; involution of orthogonal type; algebra with involution; involutions on central simple algebras; group of special similitudes Knus, M.-A., Parimala, R., Sridharan, R.: Pfaffians, central simple algebras and similitudes. Math. Z.206, 589-604 (1991) Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Rings with involution; Lie, Jordan and other nonassociative structures, Algebraic theory of quadratic forms; Witt groups and rings, Finite-dimensional division rings, Quadratic forms over general fields, Quadratic spaces; Clifford algebras, Clifford algebras, spinors, Quadratic and bilinear forms, inner products, Brauer groups of schemes, Linear algebraic groups over arbitrary fields Pfaffians, central simple algebras and similitudes

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces de Rham cohomology; Dolbeault cohomology: \(N\)-point function; hybrid cohomology classes; moduli space of Riemann surfaces A. Belopolsky, \textit{New geometrical approach to superstrings}, hep-th/9703183 [INSPIRE]. String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Supersymmetric field theories in quantum mechanics, Axiomatic quantum field theory; operator algebras, Feynman integrals and graphs; applications of algebraic topology and algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Two-loop superstrings VII. Cohomology of chiral amplitudes

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces real ruled surfaces; space of smooth real algebraic curves; Hirzebruch surface; homotheties Jean-Yves Welschinger, Courbes algébriques réelles et courbes flexibles sur les surfaces réglées de base \Bbb C\roman\Bbb P\textonesuperior , Proc. London Math. Soc. (3) 85 (2002), no. 2, 367 -- 392 (French). Rational and ruled surfaces, Families, moduli of curves (algebraic), Topology of real algebraic varieties Real algebraic curves and flexible curves on ruled surfaces of base \(\mathbb{C} \mathbb{P}^1\)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces arithmetic quotients of Hermitian symmetric spaces; arithmetic groups; moduli varieties; Burkardt quartic threefold; cubic surfaces; parameter spaces of families of abelian varieties; quintic fourfold; automorphism group B. Hunt, \textit{The Geometry of Some Special Arithmetic Quotients}, Springer Lecture Notes in Mathematics, No. 1637, Springer--Verlag, Berlin, 1996. Algebraic moduli of abelian varieties, classification, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Homogeneous spaces and generalizations, \(3\)-folds, Discrete subgroups of Lie groups, Other groups and their modular and automorphic forms (several variables) The geometry of some special arithmetic quotients

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces abstract elliptic function fields; structure of ring of meromorphisms; Riemann Hypothesis Hasse, H.: Zur Theorie der abstrakten elliptischen Funktionenkörper III. Die Struktur des Meromorphismenrings. Die Riemannsche Vermutung. J. Reine Angew. Math. \textbf{175}, 193-208 (1936) Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry Zur Theorie der abstrakten elliptischen Funktionenkörper. III: Die Struktur des Meromorphismenringes. Die Riemannsche Vermutung

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces vector bundles on a curve; ruled surfaces; moduli of vector bundles; bisecant curves; trisecant curves; Segre invariants; elementary transformation Divisors, linear systems, invertible sheaves, Rational and ruled surfaces, Vector bundles on curves and their moduli Bisecant and trisecant curves on ruled surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces symmetric product of Riemann surfaces; Jacobian; Abel-Jacobi map; higher cohomology groups Lazarsfeld, R., Cohomology on symmetric products, syzygies of canonical curves, and a theorem of kempf, No. 145, 89-97, (1993) Riemann surfaces; Weierstrass points; gap sequences, Jacobians, Prym varieties, Classical real and complex (co)homology in algebraic geometry, Compact Riemann surfaces and uniformization Cohomology on symmetric products, syzygies of canonical curves, and a theorem of Kempf

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces quadratic algebras; Hopf algebras; cohomology of flag manifolds; twisted group algebras; graded algebras; tensor products Fomin, S; Procesi, C, Fibered quadratic Hopf algebras related to Schubert calculus, J. Algebra, 230, 174-183, (2000) Quadratic and Koszul algebras, Twisted and skew group rings, crossed products, Graded rings and modules (associative rings and algebras), Grassmannians, Schubert varieties, flag manifolds Fibered quadratic Hopf algebras related to Schubert calculus

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces symmetric products of curves; Jacobian varieties; Shafarevich conjecture implies the Mordell conjecture; zeta function; Torelli theorem J.S. Milne ; '' Jacobian varieties ''. Arithmetic geometry edited by G. Cornell, J.J. Silverman, Springer-Verlag ( 1986 ). MR 861976 | Zbl 0604.14018 Jacobians, Prym varieties, Picard schemes, higher Jacobians, Families, moduli of curves (algebraic), Algebraic moduli of abelian varieties, classification Jacobian varieties

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces varieties over local fields; Chow groups; algebraic \(K\)-theory; \(p\)-divisible groups; syntomic cohomology; Dieudonné modules; rings of Witt vectors; Artinian algebras Yamazaki, T.: Formal Chow groups, p-divisible groups, and syntomic cohomology. Duke math. J. 102, 359-390 (2000) Local ground fields in algebraic geometry Formal Chow groups, \(p\)-divisible groups and syntomic cohomology.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces geometric equivalence of algebras; varieties of algebras; real-closed fields A. Berzins, ''Geometrical equivalence of algebras,''Int. J. Alg. Comput., to appear. Equational logic, Mal'tsev conditions, Ordered fields, Algebraic field extensions, Foundations of algebraic geometry Geometric equivalence of algebras.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces finite fields; function fields; hyperelliptic curves; \(K\)-groups; moments of quadratic Dirichlet \(L\)-functions; class number Andrade, J. C.; Bae, S.; Jung, H., Average values of \textit{L}-series for real characters in function fields, Res. Math. Sci., 3, (2016), 47 Curves over finite and local fields, Zeta and \(L\)-functions in characteristic \(p\), Relations with random matrices, Arithmetic theory of algebraic function fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Average values of \(L\)-series for real characters in function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Galois representations; Mordell-Weil lattices; elliptic curves; deformation theory of isolated singularities; Mordell-Weil group; Hasse zeta function; elliptic surfaces; Artin L-function; Weil height; del Pezzo surfaces; cubic forms Shioda, T.: Mordell-Weil lattices and Galois representation. I, II, III. Proc. Japan Acad., 65A, 269-271 ; 296-299 ; 300-303 (1989). Arithmetic varieties and schemes; Arakelov theory; heights, Galois theory, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Elliptic surfaces, elliptic or Calabi-Yau fibrations, General ternary and quaternary quadratic forms; forms of more than two variables, Elliptic curves over global fields, Elliptic curves Mordell-Weil lattices and Galois representation. III

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces automorphism groups; cancellation problem for function fields; function fields of general type; Zariski problem Relevant commutative algebra, Surfaces and higher-dimensional varieties, Transcendental field extensions, Algebraic functions and function fields in algebraic geometry, Group actions on varieties or schemes (quotients), Arithmetic theory of algebraic function fields Automorphism groups of ruled functions fields and a problem of Zariski

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces characteristic \(p\); \(p\)-closed rational vector fields; \(K3\) surfaces; unirational surfaces; Hirzebruch surfaces; Zariski surfaces; Enriques' surfaces; surfaces of general type; Artin invariant; quasi-elliptic surfaces M. Hirokado, Zariski surfaces as quotients of Hirzebruch surfaces by \(1\)-foliations , Yokohama Math. J., 47 (2000), 103--120. Finite ground fields in algebraic geometry, \(K3\) surfaces and Enriques surfaces, Special surfaces, Surfaces of general type, Rational and unirational varieties Zariski surfaces as quotients of Hirzebruch surfaces by 1-foliations

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces bounded symmetric domain; Satake compactification; theta functions; moduli space of arrangements; period space of K3 surfaces Matsumoto, K.: Theta functions on the classical bounded symmetric domain of type I2, 2. Proc. Japan Acad., Ser. A 67, 1--5 (1991) Other groups and their modular and automorphic forms (several variables), Homogeneous spaces and generalizations, Theta functions and abelian varieties, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) Theta functions on the classical bounded symmetric domain of type \(I_{2,2}\)