no. 15-16
Number Theory/Algebraic Geometry
P-adic weight pairings on pro-Jacobians
[Accouplements de poids P-adiques sur les pro-jacobiennes]
Présenté par : Christophe Soulé
Delbourgo, Daniel1
1 School of Mathematical Sciences, Monash University, Melbourne, Victoria 3800, Australia
Comptes Rendus. Mathématique, Tome 346 (2008) no. 15-16, pp. 819-824

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

L(E,1)≠0implies both#E(Q)<∞and#Eord<∞
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.

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

L(E,1)≠0implique#E(Q)<∞et#Eord<∞
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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.06.005

Delbourgo, Daniel 1

1 School of Mathematical Sciences, Monash University, Melbourne, Victoria 3800, Australia 
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Delbourgo, Daniel. P-adic weight pairings on pro-Jacobians. Comptes Rendus. Mathématique, Tome 346 (2008) no. 15-16, pp. 819-824. doi: 10.1016/j.crma.2008.06.005

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

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

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

[6] Hida, H. Galois representations into GL2(Zp〚X〛) attached to ordinary cusp forms, Invent. Math., Volume 85 (1986), pp. 545-613

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

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

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