Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T12:27:40.226Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-duality of Selmer groups

Published online by Cambridge University Press:  01 March 2009

TIM DOKCHITSER  and
VLADIMIR DOKCHITSER
TIM DOKCHITSER
Affiliation:
Robinson College, Cambridge CB39AN. e-mail: [email protected]
VLADIMIR DOKCHITSER
Affiliation:
Gonville & Caius College, Cambridge CB21TA. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the G-representation naturally associated to the p∞-Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 2 , March 2009 , pp. 257 - 267
Copyright
Copyright © Cambridge Philosophical Society 2008

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

REFERENCES

[1]Birch, B. J. and Stephens, N. M.The parity of the rank of the Mordell–Weil group. Topology 5 (1966), 295–299.CrossRefGoogle Scholar
[2]Bosch, S. and Liu, Q.Rational points of the group of components of a Néron model. Manuscripta Math. 98 (1999), no. 3, 275–293.CrossRefGoogle Scholar
[3]Dokchitser, T. and Dokchitser, V. On the Birch–Swinnerton-Dyer quotients modulo squares. (2006), arxiv: math.NT/0610290.Google Scholar
[4]Dokchitser, T. and Dokchitser, V. Regulator constants and the parity conjecture (2007), arxiv: 0709.2852.Google Scholar
[5]Greenberg, R.On the Birch and Swinnerton-Dyer conjecture. Invent. Math. 72, no. 2 (1983), 241–265.CrossRefGoogle Scholar
[6]Greenberg, R.Trivial zeros of p-adic L-functions. Contemp. Math. 165 (1994), 149–174.CrossRefGoogle Scholar
[7]Grothendieck, A. Modèles de Néron et monodromie, LNM 288. Séminaire de Géométrie 7, Exposé IX (Springer-Verlag, 1973).Google Scholar
[8]Guo, L.General Selmer groups and critical values of Hecke L-functions. Math. Ann. 297 no. 2 (1993), 221–233.CrossRefGoogle Scholar
[9]Kim, B. D.The parity theorem of elliptic curves at primes with supersingular reduction. Compositio Math. 143 (2007) 47–72.CrossRefGoogle Scholar
[10]MacLane, S.Homology (Springer-Verlag 1995). (Reprint of the 1975 ed.)Google Scholar
[11]Mazur, B. and Rubin, K. Finding large Selmer ranks via an arithmetic theory of local constants, arxiv: math.NT/0512085. To appear in Annals of Math.Google Scholar
[12]Milne, J. S.On the arithmetic of abelian varieties. Invent. Math. 17 (1972), 177–190.CrossRefGoogle Scholar
[13]Milne, J. S. Arithmetic duality theorems. Perspectives in Mathematics, No. 1 (Academic Press, 1986).Google Scholar
[14]Monsky, P.Generalizing the Birch–Stephens theorem. I: Modular curves. Math. Z. 221 (1996), 415–420.Google Scholar
[15]Nekovář, J.Selmer complexes. Astérisque 310 (2006).Google Scholar
[16]Silverman, J. H.Advanced Topics in the Arithmetic of Elliptic Curves. GTM 151 (Springer–Verlag 1994).CrossRefGoogle Scholar
[17]Tate, J.Number theoretic background, in: Automorphic forms, representations and L-functions, Part 2 (ed. A. Borel and W. Casselman). Proc. Symp. in Pure Math. 33 (AMS, 1979), 3–26.CrossRefGoogle Scholar