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rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) modular curve; Jacobian variety; rational point; torsion subgroup; cuspidal class number; modular unit

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) non-cuspidal rational points; quotient of the modular curve; action of the Atkin-Lehner involution; Mordell-Weil group; complex multiplication Momose, F., Rational points on the modular curves \(X_0^+(N)\), J. Math. Soc. Jpn., 39, 269-286, (1978)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) number of squares; arithmetic progression; Faltings' theorem; number of rational points; Mordell-Weil group; explicit upper bound E. Bombieri, A. Granville, J. Pintz, Squares in arithmetic progressions, \textit{Duke Math. J.} 66 (1992) 369-385.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Mordell-Faltings theorem; rational points; Mordell's conjecture; function field; Thue-Siegel-Dyson-Gel'fond theorem Vojta, P., Siegel's theorem in the compact case, \textit{Ann. Math.}, 133, 3, 509-548, (1991)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Mordell-Weil group; Abelian variety; Reduction modulo primes G. Banaszak, P. Krasoń, On arithmetic in Mordell-Weil groups, Acta Arithmetica 150 (2011), no. 4, 315--337.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) self-intersection of dualizing sheaves; elliptic curves; Arakelov theory; Arakelov-Green function; Frey curves; genus; Néron-Tate height; Jacobian Szpiro, L. 1990.Sur les propriétés numériques du dualisant relatif d'une surface arithmétique, The Grothendieck Festschrift Vol. III, 229--246. Boston: Birkhäuser.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) lattice polytopes; toric varieties; group actions on varieties; Ehrhart theory; rational polytopes; centrally symmetric polytopes; Weyl group; toric variety A. Stapledon, Equivariant Ehrhart theory, Adv. Math. 226 (2011), 3622--3654.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) finite field; finite subgroup; algebraic group; automorphism; fixed point; Smith theory Serre, J.P.: How to use finite fields for problems concerning infinite fields. In: Arithmetic, Geometry, Cryptography and Coding Theory, Volume 487 of Contemporary Mathematics, pp. 183-193. American Mathematical Society, Providence, RI (2009)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) super Riemann surfaces; dressed moduli spaces; Picard group; Picard functor; Picard variety; abelian conformal field theory; vertex operator algebra; Heisenberg algebra; conformal blocks; theta functions of higher level; infinite-dimensional Lie algebras; supercurves; Beilinson-Bernstein localization; Fock representation

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) abelian variety; inseparable; torsion point; function field; positive DOI: 10.1215/00294527-2143943

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Chabauty's theorem; number of rational points; Mordell-Weil group; hyperelliptic curve Flynn, E.V.; A flexible method for applying Chabauty's theorem; Compos. Math.: 1997; Volume 105 ,79-94.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) elliptic surfaces; Birch and Swinnerton-Dyer conjecture; elliptic curve over a function field; finite field; Drinfeld-Heegner point; Néron models; Euler systems; Drinfeld modular curves; Tate-Shafarevich group M. L. Brown, On a conjecture of Tate for elliptic surfaces over finite fields, Proc. London Math. Soc. (3) 69 (1994), no. 3, 489 -- 514.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Feynman integral; Feynman diagram; Tate motive; perturbative quantum field theory; period; oscillatory integral; Gelfand-Leray form; Connes-Kreimer theory; Radon transform; Hodge structure; noncommutative geometry; Galois group; supermanifold; Kirchhoff-Symaznik polynomial; dimensional regularization; BPHZ renormalization; Tannakian category; Grothendieck ring; monodromy; weight fibration; vanishing cycles; topological simplex; singularities; mixed Tate; tubular neighborhood; Kummer motive; Milnor fiber; motivic sheaves; normal crossings; Picard-Fuchs equation; Riemann-Hilbert correspondence; Hopf algebra; Igusa L-function Marcolli, M.: Feynman Motives. World Scientific, Singapore (2010)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) rational map; Gröbner basis; decomposition of maps; rational variety; function field Müller-Quade, J.; Steinwandt, R.; Beth, T.: An application of Gröbner bases to the decomposition of rational mappings. Gröbner bases and applications, 448-462 (1998)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) projective translation equation; flow; rational vector field; iterative functional equation; (Dixonian) elliptic function; linear partial differential equation; finite group representation; hypergeometric function; rational solution; non-singular projective flows Alkauskas, G., The projective translation equation and unramified 2-dimensional flows with rational vector fields, Aequ. Math., 89, 873-913, (2015)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Pythagorean triples; rational solutions of cubic equations; Mordell-Weil group

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) congruence subgroups; level one; subvarieties of general type; Hilbert modular variety; minimality property; automorphism group; modular function field; weights of automorphy factors; Hilbert modular group S. Tsuyumine, ``Multitensors of differential forms on the Hilbert modular variety and on its subvarieties'', Math. Ann.274 (1986) no. 4, p. 659-670 ##img## Creative Commons License BY-ND ISSN : 2429-7100 - e-ISSN : 2270-518X

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) del Pezzo variety; elliptic fibration; Mordell-Weil group; Cox ring

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Selmer group; descent; Mordell-Weil sieving; rational points; curves; Chabauty; coverings Bruin, N.: Success and challenges in determining the rational points on curves. In: Proceedings of the Tenth Algorithmic Number Theory Symposium. The Open Book Series, vol. 1.1, pp. 187-212 (2013)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Fermat quintic curve; Jacobian; dual isogeny; Mordell-Weil group Tzermias, P.: Arithmetic of cyclic quotients of the Fermat quintic, Acta arith. 84, 375-384 (1998)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Modular curves; Jacobian varieties; modular forms; Mordell-Weil group Seng-Kiat Chua & San Ling, On the rational cuspidal subgroup and the rational torsion points of \(J_0(p q)\), Proc. Am. Math. Soc.125 (1997), p. 2255-2263

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) rational point; rationally connected variety; PAC field Starr, J.; Degenerations of rationally connected varieties and PAC fields, math.AG/0602649, (2006).

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Birch--Swinnerton-Dyer conjecture; elliptic curves with complex multiplication; L-series; Mordell-Weil group; L-function B. H. Gross, ''On the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication,'' in Number Theory Related to Fermat's Last Theorem, Koblitz, N., Ed., Mass.: Birkhäuser, 1982, pp. 219-236.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) abelian variety; Mordell-Weil group; explicit descent Creutz, B.: Explicit descent in the Picard group of a cyclic cover of the projective line. In: Howe, E.W., Kedlaya K.S. (eds.) ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium, San Diego 2012. Open Book Series, vol. 1, pp. 295-315. Mathematical Science, Berkeley, California (2013)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Cremona transformation; \(\tau \)-function; Painlevé equation; rational variety; Schur function; tropical representation; universal character; Weyl group Tsuda T., Takenawa T.: Tropical representation of Weyl groups associated with certain rational varieties. Adv. Math. 221, 936--954 (2009)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) abelian variety; abelian scheme; Mordell-Weil group; support problem P. Rzonsowski, Linear relations and arithmetic on abelian schemes, Functiones et Approximatio, 52 (2015), no. 1, 83--107.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Jacobian varieties; Mordell-Weil theorem; rational points

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) \(K\)-theory; Abel-Jacobi map; Chow group; Albanese variety; intermediate Jacobian; Hodge theory Barbieri-Viale, L.; Pedrini, C.; Weibel, C., \textit{roitman's theorem for singular complex projective surfaces}, Duke Math. J., 84, 155-190, (1996)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) \(\ell\)-adic cohomology; independence of \(\ell \); grothendieck's trace formula; Lefschetz trace formula; zeta functions over finite fields; Euler-poincaré characteristic; Betti number; bloch's conductor conjecture; intersection cohomology; grothendieck's six operations; intermediate extension; Weil conjectures; Hodge polygon; Newton polygon; crystalline cohomology; Hodge filtration; coniveau filtration; alteration; Fano variety; rationally connected; Weil group; swan conductor; wild ramification; Brauer trace; log scheme; logarithmic differential forms; Čebotarev's density theorem; semisimple group; fatou's Lemma Illusie, L.: Miscellany on traces in \(\mathcall \)-adic cohomology: a survey. Japan J. Math. \textbf{1}(1), 107-136 (2006). Erratum: Japan J. Math. \textbf{2}(2), 313-314 (2007)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Weil reciprocity; class field theory; algebraic curves; function fields; residues

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Number theory; Diophantine geometry; arithmetic algebraic geometry; intersection theory; Mordell conjecture; arithmetic Nevanlinna type theory; diophantine geometry; Taniyama- Shimura-Weil conjecture; Birch-Swinnerton-Dyer conjecture; Heegner points; Mordell's conjecture; Arakelov theory; heights; diophantine approximations; integral points; minimal isogenies between 1-motives; Hasse principle S. LANG, Number Theory III, Encyclopoedia of Mathematical Sciences, Vol. 60, Springer-Verlag, 1991. Zbl0744.14012 MR1112552

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) 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).

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) inverse problem of Galois theory, algebraic function field; arithmetic fundamental group; algebraic fundamental group; special linear group; SL(2,q); Mathieu group; \(M_{12}\); \(M_{22}\) Matzat, B.H.: Zwei Aspekte konstruktiver Galoistheorie,J. Algebra 96 (1985), 499--531

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) group of rational points; cyclic; local; abelian variety; finite field

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) descent on elliptic curves; invariants of binary quartics; group of rational points; elliptic curve; algorithm; Mordell-Weil group --------, Classical invariants and \(2\)-descent on elliptic curves , J. Symbolic Comput., 31 (2001), 71-87.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Mordell-Weil rank; Mordell-Weil group; elliptic fibration; at most quadratic point of elliptic surface; estimates on heights Bremner, A.: On elliptic surfaces of Mordell-Weil rank 4,Nagoya Math. J. 102, 101--115 (1986)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Mordell's conjecture over function fields; finiteness theorems; hyperbolic manifolds; rational points; integer points; finite modulo trace; loops on complex manifolds; closed points on algebraic varieties over a finite field A. N. Parshin, ``Finiteness theorems and hyperbolic manifolds'', The Grothendieck Festschrift. A collection of articles written in honor of the 60th birthday of Alexander Grothendieck, v. III, Progr. Math., 88, Birkhaüser, Boston, MA, 1990, 163 -- 178

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Leech lattice; Mordell-Weil lattice; elliptic curves; function field of a hyperelliptic curve; minimal vectors N. D. Elkies, ''Mordell-Weil lattices in characteristic 2, II: The Leech lattice as a Mordell-Weil lattice,'' Invent. Math., vol. 128, iss. 1, pp. 1-8, 1997.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) inverse problem of Galois theory; rational variety; connectedness of the Nielsen class; rigidity of the group FRIED, M.: On reduction of the inverse Galois group problem to simple groups. Erscheint demn?chst

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) rational points; Jacobian variety; Neron-Tate height function; fixed points of a nontrivial automorphism; curves of genus \(\neq 1\)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) stable reduction; discrete valuation field; Jacobian of hyperelliptic curve; Néron model; group of components; separable field extension; Weil restriction 7. B. Edixhoven, Q. Liu and D. Lorenzini, The p-part of the group of components of a Néron model, J. Alg. Geom.5(4) (1996) 801-813.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) curve of genus two; simple 2-dimensional abelian variety; rank of Mordell-Weil group

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) formal group of the Jacobian of a complete nonsingular algebraic curve; holomorphic differentials; rational non-Weierstrass point; modular curve; cusp forms DOI: 10.2140/pjm.1993.157.241

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) hyperelliptic curve; plane quartic; rational point; zeta function; Weil bound; Serre bound E.W. Howe, K.E. Lauter, J. Top, Pointless curves of genus three and four, in \(Arithmetic, Geometry and Coding Theory (AGCT 2003)\), Séminaries & Congres, vol. 11 (société mathématique de France, Paris, 2005), pp. 125-141

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) number of integral points; rank of the group of rational points on an elliptic curve; rational points on the Jacobian variety; lower bound for the canonical height on elliptic curves; twisted Catalan curves ----,A quantitative version of Siegel's theorem, J. reine ang. Math.378 (1987), 60--100

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) 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 point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) degree of projective variety; Hilbert function; intersection theory; Bernstein-Kushnirenko theorem A. G. Khovanskii, ''Intersection theory and Hilbert function,'' Funct. Anal. Appl., vol. 45, pp. 305-315, 2011.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) curve of genus \(g\geq 2\); Coleman-Chabauty bound; upper bound; number of rational points; Jacobian; Mordell-Weil rank

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) abelian variety; the group of rational points; finite field; Littlewood-Richardson rule

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) 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.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) abelian variety; theory ACFA; group definable in a model; model theoretic stability; 1-basedness; model companion of the theory of fields with an automorphism; Manin-Mumford conjecture; projective curve; Jacobian variety

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) \(p\)-adic height pairing; Jacobian variety; arithmetic intersection theory; differentials of the third kind; de Rham cohomology R. F. Coleman and B. H. Gross, ''\(p\)-adic heights on curves,'' in Algebraic Number Theory, Adv. Stud. Pure Math. 17, Academic Press, Boston, 1989, 73--81.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) curve of genus two; rational Weierstrass point; embedding; Jacobian; formal group; theta functions Grant D.: Formal groups in genus 2. J. Reine. Angew. Math. 411, 96--121 (1990)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Siegel modular forms; cubic hypersurfaces; zero set of the Fermat ideal; Schottky modular form; Gauss map on the theta divisor; principally polarized Abelian variety; theta function; Siegel modular group; Jacobian of a generic curve; Fermat cubic; cubic forms Mccrory, C.; Shifrin, T.; Varley, R.: Siegel modular forms generated by invariants of cubic hypersurfaces. J. algebraic geom. 4, 527-556 (1995)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) pro-\(\ell \) fundamental group; rational function field; maximum pro-\(\ell \) extension; absolute Galois group; Frobenius automorphism

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Brauer group; Galois cohomology; rationally connected variety; function field

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) \(abc\) conjecture; function field; Kobayashi hyperbolicity; Nevanlinna theory; rational points [N] J. Noguchi,Nevanlinna-Cartan theory over function fields and a Diophantine equation, Journal für die reine und angewandte Mathematik487 (1997), 61--83.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) elliptic curve with complex multiplication; \(\mathbb Z_p\)-extension; imaginary quadratic number field; Leopoldt conjecture; Selmer group; Mordell-Weil group; Birch-Swinnerton-Dyer conjecture B. Perrin-Riou, Groupe de Selmer d'une courbe elliptique à multiplication complexe, Compos. Math. 43 (1981), 387--417.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) finite field; Hasse-Weil bound; permutation; rational function

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) curve; rational point; zeta function; Weil bound; Serre bound; Oesterlé bound Howe, Improved upper bounds for the number of points on curves over finite fields, Ann. Inst. Fourier (Grenoble) 53 pp 1677-- (2003)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) toric variety; Weil divisor; Cartier divisor; convexity; Picard number; polytope; polyhedron; sheaf cohomology; Hizrebruch-Jung continued fraction; Gröbner fan; McKay correspondence; Rees algebra; multiplier ideal; Hirzebruch-Riemann-Roch theorem; Chow ring; intersection cohomology; invariant theory; GKZ cone; secondary fan; spectral sequence D. A. Cox, J. B. Little, and H. K. Schenck, \textit{Toric varieties}, Graduate Studies in Mathematics, 124, American Mathematical Society, Providence, RI, 2011.Zbl 1223.14001 MR 2810322

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) finite fields; algebraic function fields; algebraic curves; Riemann-Roch theorem; rational places; coding theory; algebraic-geometry codes; function-field codes; elliptic and hyperelliptic curve cryptography; McEliece and Niederreiter cryptosystems; frameproof codes H. Niederreiter and C.P. Xing. \textit{Algebraic geometry in coding theory and cryptography}. Princeton University Press, Princeton, NJ (2009).

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Weil reciprocity law; class field theory; algebraic curves; function fields

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) noncommutative rational function; invariant field; group representation; positive rational function

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Eisenstein ideals; Mordell-Weil groups; Eisenstein quotient of jacobian variety; modular curves; Galois covering; cusps Kamienny, S., Rational points on modular curves and abelian varieties, J. Reine Angew. Math., 359, 174-187, (1985)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) abelian variety; local-global principle; Mordell-Weil group; 1-motive Jossen, P.: Detecting linear dependence on a simple Abelian variety. Comment. Math. Helv., to appear, 24~p

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) curves over number fields; diophantine equations; symmetric power of algebraic curves; Mordell conjecture; Jacobian variety; geometric gap principle; arithmetic gap priciple; gap principle for abelian varieties; quadratic point Joseph H. Silverman, Rational points on symmetric products of a curve, Amer. J. Math. 113 (1991), no. 3, 471 -- 508.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) group of automorphisms; birational splitting theorem for the Albanese map; Albanese variety; meromorphic function field

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) curves of genus greater than one; computational aspects; number of points; Jacobian; finite fields; Mordell-Weil group; construction of curves Poonen, Bjorn, Computational aspects of curves of genus at least \(2\). Algorithmic number theory, Talence, 1996, Lecture Notes in Comput. Sci. 1122, 283-306, (1996), Springer, Berlin

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) finite field; rational point of algebraic variety A. Cafure and G. Matera, \textit{Fast computation of a rational point of a variety over a finite field}, Math. Comp. \textbf{75} (2006), no.~256,2049-2085.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) absolute irreducibility; finite field; Hasse-Weil bound; permutation; rational function

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Algebraic curves; rational points; Chabauty method; Jacobian; Mordell-Weil sieve Stoll, M., Rational points on curves, J. Théor. Nombres Bordeaux, 23, 1, 257-277, (2011)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) elliptic curves; Mordell-Weil group; Thue-Siegel-Roth theorem; cubic form; rank of group of rational points DOI: 10.1007/BF01389220

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Cassels-Tate pairing; Albanese-Albanese definition; Weil pairing definition; genus \(g\) curve; Shafarevich-Tate group; abelian variety; Jacobian; hyperelliptic curves; local densities; Shimura curves; Brauer groups B. Poonen and M. Stoll, \textit{The Cassels--Tate pairing on polarized abelian varieties}, Ann. of Math. (2), 150 (1999), pp. 1109--1149.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) real function fields; real holomorphy ring; valuation rings; formally real residue field; regular projective model; rational point; real prime divisors; signatures of higher level; sums of n-th powers; local-global principle; weak isotropy; quadratic forms; Henselizations Schülting, H. W.: The binary class group of the real holomorphy ring. (1986)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) conjugacy classes; representation theory of finite groups of Lie type; connected reductive algebraic group; Frobenius morphism; finite group of F-fixed points; BN-pair; Weyl group; Borel subgroup; maximal torus; intersections of parabolic subgroups; virtual representation; generalized character; \(\ell \)-adic cohomology; flag variety; geometric conjugacy; Jordan decomposition; unipotent elements; Deligne-Lusztig characters; irreducible characters; regular characters; Gelfand-Graev representation; cuspidal characters; unipotent characters; intersection cohomology; Springer correspondence; representations of Coxeter groups; Dynkin diagrams; bibliography; exceptional groups; character tables R.\ W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, John Wiley, New York, 1985.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Tate module; abelian variety; function field; Galois group; transcendence degree

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) rational point; elementary theory of algebraic PAC fields; maximal PRC fields; absolute Galois group Jarden, M., The algebraic nature of the elementary theory of PRC fields, Manuscripta math., 60, 463-475, (1988)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Diophantine geometry; arithmetic and algebraic geometry; abelian and Jacobian varieties; Mordell-Weil group; rank

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) commensurability; abelian variety; torus; reduction map; Mordell-Weil group

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Noether's problem; rational field extension; purely transcendental; \(p\)-group; meta-abelian group; action; field automorphism; rational function field Kunyavskiĭi, B. E.: The Bogomolov multiplier of finite simple groups, Cohomological and geometric approaches to rationality problems, 209217, Progr. Math., 282, Birkhäuser Boston, Inc., Boston, MA, 2010

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) undecidable positive existential theory; rational function field; polynomial ring Pheidas, T., and K. Zahidi, ''Undecidable existential theories of polynomial rings and function fields'', Communications in Algebra , vol. 27 (1999), pp. 4993--5010.

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rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) exact sequence; zeta function; elliptic curve; Mordell-Weil rank; abelian variety Ulmer, Douglas, Curves and Jacobians over function fields.Arithmetic geometry over global function fields, Adv. Courses Math. CRM Barcelona, 283-337, (2014), Birkhäuser/Springer, Basel

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) finite field; algebraic variety; rational point; Smith normal form; exponent matrix

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Shimura variety; orthogonal group; Siegel-Weil formula; kudla program; Whittaker function; special cycle; Green current

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) singular plane quartic curve; function field; Galois group; Galois point Miura K.: Field theory for function fields of singular plane quartic curves. Bull. Aust. Math. Soc. 62, 193--204 (2000)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Veronese variety; Segre variety; minimal degree variety; rational point; finite field Nullstellensatz; characteristic \(p\) E. Ballico and A. Cossidente, On the finite field Nullstellensatz, Australasian Journal of Combinatorics, 21 (2000), 57--60.

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rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Fano threefold; rational variety; intermediate jacobian; non-closed field; abelian surface

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) symmetric space associated to the semisimple real algebraic group; torsionfree arithmetic subgroup; locally symmetric space; Hilbert modular variety; (2n)-dimensional real hyperbolic space; \(L_ 2\)-cohomology; middle intersection homology; neighborhood of a boundary point; warped product S. Zucker : L2-cohomology and intersection homology of locally symmetric varieties . Proc. Symposia Pure Math. 40, Part 2 (1983) 675-680. Providence: AMS.

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rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Lang-Weil estimates; number of rational points; variety over a finite field; Galois stratification; ring cover; Artin symbol M. D. Fried, D. Haran and M. Jarden, Effective counting of the points of definable sets over finite fields, Israel J. Math. 85(1--3) (1994), 103--133.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) rational point; product theorem; abelian variety; abelian subvariety; Mordell's conjecture; ample symmetric invertible sheaf; Néron-Tate height pairing; Faltings' product theorem

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) real function fields; real holomorphy ring; valuation rings; formally real residue field; regular projective model; rational point; real prime divisors; signatures of higher level; sums of n-th powers; Local-Global- Principle; weak isotropy; quadratic forms; Henselizations SCHÜLTING, H.W.: Prime divisors on real varieties and valuation theory. J. Alg.98, 499-514 (1986)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) function field analogue of Mordell's conjecture; approximations of rational numbers by rationals; Grothendieck-Riemann-Roch theorem [V2] P. Vojta: Mordell's conjecture over function fields. Inv. Math.,98, 115--138 (1989)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Hecke zeta-function; non-abelian L-functions; scalar product of; L- functions; ideals with equal norms; integral points on norm form; varieties; meromorphic continuation; norm form equations; equidistribution of prime ideals; formal Euler product; absolute Weil group; algebraic number field; representations; virtual characters; Frobenius class; Größencharakteren B. Z. Moroz, \textit{Analytic Arithmetic in Algebraic Number Fields} (Springer, Berlin, 1986), Lect. Notes Math. 1205.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) rational elliptic surfaces; Mordell-Weil group; Zariski \(N\)-plet; dihedral covers Bannai, S.; Tokunaga, H.-o., Geometry of bisections of elliptic surfaces and Zariski \textit{N}-plets for conic arrangements, Geom. dedic., 178, 219-237, (2015)

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) elliptic curves over function fields; Mordell-Weil lattices; \(L\)-function of an elliptic curve over a function field T. Shioda, Some remarks on elliptic curves over function fields , Astérisque 209 (1992), 12, 99-114.

rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) curve; finite field; Jacobian; abelian variety; Fermat curve; Frobenius; Verschiebung; group scheme; Ekedahl-Oort type; de Rham cohomology; Dieudonné module