×

Curves and Jacobians over function fields. (English) Zbl 1384.11077

Bars, Francesc (ed.) et al., Arithmetic geometry over global function fields. Selected notes based on the presentations at five advanced courses on arithmetic geometry at the Centre de Recerca Matemàtica, CRM, Barcelona, Spain, February 22 – March 5 and April 6–16, 2010. Basel: Birkhäuser/Springer (ISBN 978-3-0348-0852-1/pbk; 978-3-0348-0853-8/ebook). Advanced Courses in Mathematics – CRM Barcelona, 281-337 (2014).
From the text: These notes originated in a 12-hour course of lectures given at the Centre de Recerca Matem‘atica in February 2010. The aim of the course was to explain results on curves and their Jacobians over function fields, with emphasis on the group of rational points of the Jacobian, and to explain various constructions of Jacobians with large Mordell-Weil rank.
For the entire collection see [Zbl 1305.11001].

MSC:

11G35 Varieties over global fields
11G50 Heights
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14H40 Jacobians, Prym varieties
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Artin. Supersingular K3 surfaces. Ann. Sci. ´Ecole Norm. Sup. (4), 7:543- 567 (1975), 1974. · Zbl 0322.14014
[2] M. Artin. Lipman’s proof of resolution of singularities for surfaces. In Arithmetic Geometry (Storrs, Conn., 1984), pages 267-287. Springer, New York, 1986. · Zbl 0602.14011
[3] M. Artin. N´eron models. In Arithmetic Geometry (Storrs, Conn., 1984), pages 213-230. Springer, New York, 1986. · Zbl 0603.14028
[4] M. Artin and H.P.F. Swinnerton-Dyer. The Shafarevich-Tate conjecture for pencils of elliptic curves on K3 surfaces. Invent. Math., 20:249-266, 1973. · Zbl 0289.14003
[5] M. Artin and G. Winters. Degenerate fibres and stable reduction of curves. Topology, 10:373-383, 1971. · Zbl 0196.24403
[6] L. B˘adescu. Algebraic Surfaces. Universitext. Springer-Verlag, New York, 2001. Translated from the 1981 Romanian original by Vladimir Ma¸sek and revised by the author.
[7] W. Bauer. On the conjecture of Birch and Swinnerton-Dyer for abelian varieties over function fields in characteristic p > 0. Invent. Math., 108:263-287, 1992. · Zbl 0807.14014
[8] L. Berger. Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields. J. Number Theory, 128:3013-3030, 2008. · Zbl 1204.11098
[9] S. Bosch, W. L¨utkebohmert, and M. Raynaud. N´eron Models, vol. 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1990. · Zbl 0705.14001
[10] P. Colmez and J.-P. Serre, eds. Correspondance Grothendieck-Serre. Documents Math´ematiques (Paris) [Mathematical Documents (Paris)], 2. Soci´et´e Math´ematique de France, Paris, 2001. · Zbl 0986.01019
[11] B. Conrad. Chow’s K/k-image and K/k-trace, and the Lang-N´eron theorem. Enseign. Math. (2), 52:37-108, 2006. · Zbl 1133.14028
[12] D.A. Cox. The Noether-Lefschetz locus of regular elliptic surfaces with section and pg≥ 2. Amer. J. Math., 112:289-329, 1990. · Zbl 0721.14017
[13] P. Deligne. La conjecture de Weil, II. Inst. Hautes ´Etudes Sci. Publ. Math., (52):137-252, 1980. · Zbl 0456.14014
[14] G. Faltings. Finiteness theorems for abelian varieties over number fields. In Arithmetic Geometry (Storrs, Conn., 1984), pages 9-27. Springer, New York, 1986. Translated from the German original [Invent. Math. 73 (1983), no. 3, 349-366; ibid. 75 (1984), no. 2, 381; MR 85g:11026ab] by Edward Shipz. · Zbl 0588.14026
[15] M. Fontana. Sur le plongement projectif des surfaces d´efinies sur un corps de base fini. Compositio Math., 31:1-22, 1975. · Zbl 0341.14009
[16] W.J. Gordon. Linking the conjectures of Artin-Tate and Birch-SwinnertonDyer. Compositio Math., 38:163-199, 1979. · Zbl 0425.14003
[17] B.H. Gross. Local heights on curves. In Arithmetic Geometry (Storrs, Conn., 1984), pages 327-339. Springer, New York, 1986. · Zbl 0605.14027
[18] A. Grothendieck. Fondements de la g´eom´etrie alg´ebrique. [Extraits du S´eminaire Bourbaki, 1957-1962.]. Secr´etariat math´ematique, Paris, 1962. · Zbl 0239.14002
[19] A. Grothendieck. Le groupe de Brauer, III. Exemples et compl´ements. In Dix expos´es sur la cohomologie des sch´emas, pages 88-188. North-Holland, Amsterdam, 1968. · Zbl 0198.25901
[20] R. Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics, 52. Springer-Verlag, New York, 1977. · Zbl 0367.14001
[21] M. Hindry and A. Pacheco. Sur le rang des jacobiennes sur un corps de fonctions. Bull. Soc. Math. France, 133:275-295, 2005. · Zbl 1082.11043
[22] L. Illusie. Complexe de de Rham-Witt. In Journ´ees de G´eom´etrie Alg´ebrique de Rennes (Rennes, 1978), Vol. I, volume 63 of Ast´erisque, pages 83-112. Soc. Math. France, Paris, 1979. · Zbl 0446.14008
[23] L. Illusie. Finiteness, duality, and K¨unneth theorems in the cohomology of the de Rham-Witt complex. In Algebraic Geometry (Tokyo/Kyoto, 1982), volume 1016 of Lecture Notes in Math., pages 20-72. Springer, Berlin, 1983. · Zbl 0538.14013
[24] K. Kato and F. Trihan. On the conjectures of Birch and Swinnerton-Dyer in characteristic p > 0. Invent. Math., 153:537-592, 2003. · Zbl 1046.11047
[25] N.M. Katz. Moments, Monodromy, and Perversity: a Diophantine Perspective, volume 159 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2005. · Zbl 1079.14025
[26] S.L. Kleiman. The Picard scheme. In Fundamental Algebraic Geometry, volume 123 of Math. Surveys Monogr., pages 235-321. Amer. Math. Soc., Providence, RI, 2005.
[27] K. Kramer. Two-descent for elliptic curves in characteristic two. Trans. Amer. Math. Soc., 232:279-295, 1977. · Zbl 0327.14007
[28] S. Lichtenbaum. Curves over discrete valuation rings. Amer. J. Math., 90:380- 405, 1968. · Zbl 0194.22101
[29] S. Lichtenbaum. Duality theorems for curves over p-adic fields. Invent. Math., 7:120-136, 1969. · Zbl 0186.26402
[30] S. Lichtenbaum. Zeta functions of varieties over finite fields at s = 1. In Arithmetic and Geometry, Vol. I, volume 35 of Progr. Math., pages 173-194. Birkh¨auser Boston, Boston, MA, 1983. · Zbl 0567.14015
[31] C. Liedtke. A note on non-reduced Picard schemes. J. Pure Appl. Algebra, 213:737-741, 2009. · Zbl 1156.14031
[32] J. Lipman. Desingularization of two-dimensional schemes. Ann. of Math. (2), 107:151-207, 1978. · Zbl 0349.14004
[33] Q. Liu. Algebraic Geometry and Arithmetic Curves, volume 6 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2002. Translated from the French by Reinie Ern´e, Oxford Science Publications. · Zbl 0996.14005
[34] Q. Liu, D. Lorenzini, and M. Raynaud. N´eron models, Lie algebras, and reduction of curves of genus one. Invent. Math., 157:455-518, 2004. · Zbl 1060.14037
[35] Q. Liu, D. Lorenzini, and M. Raynaud. On the Brauer group of a surface. Invent. Math., 159:673-676, 2005. · Zbl 1077.14023
[36] Ju.I. Manin. Cyclotomic fields and modular curves. Uspekhi Mat. Nauk, 26:7- 71, 1971. · Zbl 0241.14014
[37] J.S. Milne. Elements of order p in the Tate-ˇSafareviˇc group. Bull. London Math. Soc., 2:293-296, 1970. · Zbl 0205.50801
[38] J.S. Milne. On a conjecture of Artin and Tate. Ann. of Math. (2), 102:517- 533, 1975. · Zbl 0343.14005
[39] J.S. Milne. ´Etale Cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1980. · Zbl 0433.14012
[40] J.S. Milne. Arithmetic Duality Theorems, volume 1 of Perspectives in Mathematics. Academic Press Inc., Boston, MA, 1986. · Zbl 0613.14019
[41] J.S. Milne. Jacobian varieties. In Arithmetic Geometry (Storrs, Conn., 1984), pages 167-212. Springer, New York, 1986. · Zbl 0604.14018
[42] J.S. Milne. Values of zeta functions of varieties over finite fields. Amer. J. Math., 108:297-360, 1986. · Zbl 0611.14020
[43] L. Moret-Bailly. Pinceaux de vari´et´es ab´eliennes. Ast´erisque, 129:266, 1985. · Zbl 0595.14032
[44] D. Mumford. Lectures on Curves on an Algebraic Surface. With a section by G.M. Bergman. Annals of Mathematics Studies, No. 59. Princeton University Press, Princeton, N.J., 1966. · Zbl 0187.42701
[45] D. Mumford and T. Oda. Algebraic Geometry II. In preparation. · Zbl 1325.14001
[46] N. Nygaard and A. Ogus. Tate’s conjecture for K3 surfaces of finite height. Ann. of Math. (2), 122:461-507, 1985. · Zbl 0591.14005
[47] T. Occhipinti. Mordell-Weil groups of large rank in towers. ProQuest LLC, Ann Arbor, MI, 2010. Ph.D. Thesis, The University of Arizona.
[48] C. Pomerance and D. Ulmer. On balanced subgroups of the multiplicative group, in Number theory and related fields, volume 43 of Springer Proc. Math. Stat., pages 253-270, Springer, New York, 2013. · Zbl 1332.11089
[49] R. Pries and D. Ulmer. Arithmetic of abelian varieties in Artin-Schreier extensions, preprint, 2013, arxiv:1305.5247 · Zbl 1410.11061
[50] M. Raynaud. Sp´ecialisation du foncteur de Picard. Inst. Hautes ´Etudes Sci. Publ. Math., 38:27-76, 1970. · Zbl 0207.51602
[51] P. Schneider. Zur Vermutung von Birch und Swinnerton-Dyer ¨uber globalen Funktionenk¨orpern. Math. Ann., 260:495-510, 1982. · Zbl 0509.14022
[52] C. Schoen. Varieties dominated by product varieties. Internat. J. Math., 7:541-571, 1996. · Zbl 0907.14002
[53] J.-P. Serre. Sur la topologie des vari´et´es alg´ebriques en caract´eristique p. In Symposium internacional de topolog´ıa algebraica, International Symposium on Algebraic Topology, pages 24-53. Universidad Nacional Aut´onoma de M´exico and UNESCO, Mexico City, 1958. · Zbl 0098.13103
[54] J.-P. Serre. Local Fields, volume 67 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1979. Translated from the French by Marvin Jay Greenberg. · Zbl 0423.12016
[55] Th´eorie des intersections et th´eor‘eme de Riemann-Roch. Lecture Notes in Mathematics, Vol. 225. Springer-Verlag, Berlin, 1971. S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1966-1967 (SGA 6). Dirig´e par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J.P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J.-P. Serre.
[56] T. Shioda. An explicit algorithm for computing the Picard number of certain algebraic surfaces. Amer. J. Math., 108:415-432, 1986. · Zbl 0602.14033
[57] T. Shioda. Mordell-Weil lattices and sphere packings. Amer. J. Math., 113:931-948, 1991. · Zbl 0756.14010
[58] T. Shioda. Mordell-Weil lattices for higher genus fibration over a curve. In New Trends in Algebraic Geometry (Warwick, 1996), volume 264 of London Math. Soc. Lecture Note Ser., pages 359-373. Cambridge Univ. Press, Cambridge, 1999. · Zbl 0947.14012
[59] M. Szydlo. Elliptic fibers over non-perfect residue fields. J. Number Theory, 104:75-99, 2004. · Zbl 1041.14001
[60] J.T. Tate. Algebraic cycles and poles of zeta functions. In Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), pages 93-110. Harper & Row, New York, 1965. · Zbl 0213.22804
[61] J.T. Tate. Endomorphisms of abelian varieties over finite fields. Invent. Math., 2:134-144, 1966. · Zbl 0147.20303
[62] J.T. Tate. On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. In S´eminaire Bourbaki, Vol. 9, Exp. No. 306, pages 415-440. Soc. Math. France, Paris, 1966. · Zbl 0199.55604
[63] J.T. Tate. Conjectures on algebraic cycles in -adic cohomology. In Motives (Seattle, WA, 1991), volume 55 of Proc. Sympos. Pure Math., pages 71-83. Amer. Math. Soc., Providence, RI, 1994. · Zbl 0814.14009
[64] D. Ulmer. p-descent in characteristic p. Duke Math. J., 62:237-265, 1991. · Zbl 0742.14028
[65] D. Ulmer. On the Fourier coefficients of modular forms, II. Math. Ann., 304:363-422, 1996. · Zbl 0856.11022
[66] D. Ulmer. Elliptic curves with large rank over function fields. Ann. of Math. (2), 155:295-315, 2002. · Zbl 1109.11314
[67] D. Ulmer. Geometric non-vanishing. Invent. Math., 159:133-186, 2005. · Zbl 1105.11017
[68] D. Ulmer. L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields. Invent. Math., 167:379-408, 2007. · Zbl 1110.11019
[69] D. Ulmer. Explicit points on the Legendre curve, J. Number Theory, 136:165- 194, 2014. · Zbl 1297.11055
[70] D. Ulmer. Elliptic curves over function fields. In Arithmetic of L-functions (Park City, UT, 2009), volume 18 of IAS/Park City Math. Ser., pages 211- 280. Amer. Math. Soc., Providence, RI, 2011. · Zbl 1323.11037
[71] D. Ulmer. On Mordell-Weil groups of Jacobians over function fields, J. Inst. Math. Jussieu, 12:1-29, 2013. · Zbl 1318.14025
[72] D. Ulmer and Y.G. Zarhin. Ranks of Jacobians in towers of function fields. Math. Res. Lett., 17:637-645, 2010. · Zbl 1256.14024
[73] Ju.G. Zarhin. Abelian varieties in characteristic p. Mat. Zametki, 19:393-400, 1976. · Zbl 0342.14011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.