<i>P</i>-adic weight pairings on pro-Jacobians

Daniel Delbourgo

doi:10.1016/j.crma.2008.06.005

Number Theory/Algebraic Geometry

P-adic weight pairings on pro-Jacobians
[Accouplements de poids P-adiques sur les pro-jacobiennes]

Présenté par : Christophe Soulé

Daniel Delbourgo ¹

¹ School of Mathematical Sciences, Monash University, Melbourne, Victoria 3800, Australia

Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 819-824.

Résumé
Abstract

Soit E une courbe elliptique définie sur le corps des nombres rationnels. Nous proposons une méthode pour démontrer l'énoncé

L (E, 1) \neq 0 implique # E (Q) < \infty et #_{E}^{ord} < \infty

en utilisant des formes modulaires p-adiques, c'est-à-dire sans la théorie d'Iwasawa. La démonstration utilise une version des éléments-zêta à coefficients Λ-adiques.

Let E denote an elliptic curve defined over the rational numbers. We outline a method of proving the statement

L (E, 1) \neq 0 implies both # E (Q) < \infty and #_{E}^{ord} < \infty

using properties of p-adic modular forms, i.e. no Iwasawa theory whatsoever. The proof employs a version of Kato's zeta-elements with Λ-adic coefficients.

Reçu le : 2008-03-28
Accepté le : 2008-06-10
Publié le : 2008-07-21

DOI : 10.1016/j.crma.2008.06.005

Affiliations des auteurs :

Daniel Delbourgo ¹

¹ School of Mathematical Sciences, Monash University, Melbourne, Victoria 3800, Australia

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     author = {Daniel Delbourgo},
     title = {\protect\emph{P}-adic weight pairings on {pro-Jacobians}},
     journal = {Comptes Rendus. Math\'ematique},
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     doi = {10.1016/j.crma.2008.06.005},
     language = {en},
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Daniel Delbourgo. P-adic weight pairings on pro-Jacobians. Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 819-824. doi : 10.1016/j.crma.2008.06.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.06.005/

Bibliographie
Cité par

[1] D. Delbourgo Λ-adic Euler characteristics of elliptic curves, Documenta Math., Volume volume in honour of J.H. Coates' 60th birthday (2006), pp. 301-323

[2] D. Delbourgo Elliptic Curves and Big Galois Representations, London Mathematical Society Lecture Note Series, vol. 356, Cambridge University Press, 2008

[3] D. Delbourgo, On the divisibility of Selmer into the improved p-adic L-function, in preparation

[4] D. Delbourgo; P. Smith Kummer theory for big Galois representations, Math. Proc. Cambridge Philos. Soc., Volume 142 (2007), pp. 205-217

[5] R. Greenberg; G. Stevens p-adic L-functions and p-adic periods of modular forms, Invent. Math., Volume 111 (1993), pp. 401-447

[6] H. Hida Galois representations into ${GL}_{2} (Z_{p} 〚 X 〛)$ attached to ordinary cusp forms, Invent. Math., Volume 85 (1986), pp. 545-613

[7] H. Hida Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4), Volume 19 (1986), pp. 231-273

[8] H. Hida A p-adic measure attached to the zeta-functions associated with two elliptic modular forms I, Invent. Math., Volume 79 (1985), pp. 159-195

[9] J. Nekovář; A. Plater On the parity of ranks of Selmer groups, Asian J. Math., Volume 4 (2000) no. 2, pp. 437-497

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