Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T02:59:42.391Z Has data issue: false hasContentIssue false
  • English
  • Français

The $l$-parity conjecture for abelian varieties over function fields of characteristic $p>0$

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  10 March 2014

Fabien Trihan  and
Seidai Yasuda
Fabien Trihan
Affiliation:
School of Engineering, Computing and Mathematics, University of Exeter, EX4 4QF, UK email [email protected]
Seidai Yasuda
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $A/K$ be an abelian variety over a function field of characteristic $p>0$ and let $\ell $ be a prime number ($\ell =p$ allowed). We prove the following: the parity of the corank $r_\ell $ of the $\ell $-discrete Selmer group of $A/K$ coincides with the parity of the order at $s=1$ of the Hasse–Weil $L$-function of $A/K$. We also prove the analogous parity result for pure $\ell $-adic sheaves endowed with a nice pairing and in particular for the congruence Zeta function of a projective smooth variety over a finite field. Finally, we prove that the full Birch and Swinnerton-Dyer conjecture is equivalent to the Artin–Tate conjecture.

Keywords

elliptic curvesSelmer groupsparity conjecture

MSC classification

Secondary: 11G20: Curves over finite and local fields 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 4 , April 2014 , pp. 507 - 522
Copyright
© The Author(s) 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abe, T. and Caro, D., Theory of weights in p-adic cohomology, Preprint (2013), arXiv:1303.0622.Google Scholar
Berthelot, P., Breen, L. and Messing, W., Théorie de Dieudonné Cristalline II, Lecture Notes in Mathematics, vol. 930 (Springer, 1982).Google Scholar
Coates, J., Fukaya, T., Kato, K. and Sujatha, R., Root numbers, Selmer groups, and non-commutative Iwasawa theory, J. Algebraic Geom. 19 (2010), 19–97.Google Scholar
Česnavičius, K., The p-parity conjecture for elliptic curves with a p-isogeny. Preprint (2012),arXiv:1207.0431v2.Google Scholar
Crew, R., Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve, Ann. Sci. Éc. Norm. Supér. (4) 31 (1998), 717–763.Google Scholar
Deligne, P., La conjecture de Weil : II, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137–252.Google Scholar
Dokchitser, T. and Dokchitser, V., Parity of ranks for elliptic curves with a cyclic isogeny, J. Number Theory 128 (2008), 662–679.CrossRefGoogle Scholar
Dokchitser, T. and Dokchitser, V., Regulator constants and the parity conjecture, Inv. Math. 178 (2009), 23–71.CrossRefGoogle Scholar
Dokchitser, T. and Dokchitser, V., On the Birch–Swinnerton-Dyer quotients modulo squares, Ann. of Math. (2) 172 (2010), 567–596.CrossRefGoogle Scholar
Dokchitser, T. and Dokchitser, V., Root numbers and parity of ranks of elliptic curves, J. Reine Angew. Math. 2011 (658) (2011), 39–64.Google Scholar
Grothendieck, A. and Dieudonné, J., Eléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Etude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Études Sci. 8 (1961), 5–222.Google Scholar
Etesse, J.-Y. and Le Stum, B., Fonctions L associés aux F-isocristaux surconvergents I. Interprétation cohomologique, Math. Ann. 296 (1993), 557–576.CrossRefGoogle Scholar
Grothendieck, A., Le groupe de Brauer III, in Dix expos’es sur la cohomologie des schémas (North Holland, 1968).Google Scholar
Hartshorne, R., Algebraic geometry (Springer, New York, 1977), corrected 6th printing, 1993.CrossRefGoogle Scholar
Kato, K. and Trihan, F., On the conjecture of Birch and Swinnerton-Dyer in characteristic p > 0, Invent. Math. 153 (2003), 537–592.Google Scholar
Katz, N. and Messing, W., Some consequences for the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77.CrossRefGoogle Scholar
Kedlaya, K., Full faithfulness for overconvergent F-isocrystals, Geometric Aspects of Dwork Theory, vol. II (de Gruyter, Berlin, 2004), 819–835.Google Scholar
Kedlaya, K., Fourier transform and p-adic Weil II, Compositio Math. 142 (2006), 1426–1450.Google Scholar
Kiehl, R. and Weissauer, R., Weil conjectures, perverse sheaves and l-adic Fourier transform, Ergebnisse de Mathematik und ihere Grenzgebiete vol. 42 (Springer, Berlin, 2001).Google Scholar
Kim, B. D., The parity conjecture for elliptic curves at supersingular reduction primes, Compositio Math. 143 (2007), 47–72.Google Scholar
Kim, B. D., The symmetric structure of the plus/minus Selmer groups of elliptic curves over totally real fields and the parity conjecture, J. Number Theory 129 (2009), 1149–1160.Google Scholar
Lafforgue, L., Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), 1–242.Google Scholar
Liu, Q., Algebraic geometry and arithmetic curves, Transl. by Reinie Ern. (English) Oxford Graduate Texts in Mathematics, vol. 6 (Oxford University Press, Oxford, 2006).Google Scholar
Liu, Q., Lorenzini, D. and Raynaud, M., On the Brauer group of a surface, Invent. Math. 159 (2005), 673–676.Google Scholar
Matsuda, S. and Trihan, F., Image directe supérieure et unipotence, J. Reine Angew. Math. 2004 (2004), 47–54.Google Scholar
Milne, J., Abelian varieties, in Proc. conf. on arithmetic geometry, Storrs, 1984 (Springer, New York, 1984), 103–150.Google Scholar
Nekovář, J., On the parity of ranks of Selmer groups II, C. R. Acad. Sci. Paris, Ser. I 332 (2001), 99–104.Google Scholar
Nekovář, J., Selmer complexes, Astérisque 310 (2006).Google Scholar
Nekovář, J., On the parity of ranks of Selmer groups III, Doc. Math. 12 (2007), 243–274; Erratum: Doc. Math. 14 (2009), 191–194.Google Scholar
Nekovář, J., On the parity of ranks of Selmer groups IV, Compositio Math. 145 (2009), 1351–1359; with an appendix by J.-P. Wintenberger.Google Scholar
Nekovář, J., Some consequences of a formula of Mazur and Rubin for arithmetic local constants, Algebra Number Theory 7 (2013), 1101–1120.Google Scholar
Oda, T., The first de Rham cohomology group and Dieudonné modules, Ann. Sci. Éc. Norm. Supér. (4) (1969), 63–135.Google Scholar
Schneider, P., Zur Vermutung von Birch und Swinnerton-Dyer über globalen Funktionskörpern., Math. Ann. 260 (1982), 495–510.CrossRefGoogle Scholar
Tate, J., Conjectures on algebraic cycles in ℓ-adic cohomology, in Motives. Proceedings of the Summer Research Conference on Motives, University of Washington, Seattle, WA, USA, July 20–August 2, 1991, Proceedings of Symposia in Pure Mathematics, vol. 55, eds Jannsen, U. et al. (American Mathematical Society, Providence, RI, 1994), 71–83; Pt. 1.Google Scholar
Trihan, F. and Wuthrich, C., Parity conjectures for elliptic curves over global fields of positive characteristic, Compositio Math. 147 (2011), 1105–1128.Google Scholar
Ulmer, D., Curves and Jacobians over function fields, Preprint (2012),http://people.math.gatech.edu/∼ulmer/research/preprints/C.pdf.Google Scholar