On anticyclotomic variants of the $p$-adic Birch and Swinnerton-Dyer conjecture

Agboola, Adebisi; Castella, Francesc

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.1, pp. 629-658.

Nous formulons des analogues de la conjecture de Birch et Swinnerton-Dyer pour les fonctions $L$ $p$-adiques de Bertolini, Darmon et Prasanna attachées aux courbes elliptiques $E/\mathbf{Q}$ en leurs places de bonne réduction ordinaire. En utilisant la théorie d’Iwasawa, nous prouvons ensuite, sous des hypothèses faibles, l’une des inégalités prédites par la partie rang de nos conjectures, ainsi que la formule prédite pour la valeur du premier terme non nul dans le développement limité, à une unité $p$-adique près.

Nos conjectures sont très étroitement liées aux conjectures du type Birch et Swinnerton-Dyer formulées par Bertolini et Darmon en 1996 pour les distributions de Heegner, et comme application de nos résultats, nous obtenons également la preuve d’une inégalité dans la partie rang de leurs conjectures.

We formulate analogues of the Birch and Swinnerton-Dyer conjecture for the $p$-adic $L$-functions of Bertolini, Darmon, and Prasanna attached to elliptic curves $E/\mathbf{Q}$ at primes $p$ of good ordinary reduction. Using Iwasawa theory, we then prove, under mild hypotheses, one of the inequalities predicted by the “rank part” of our conjectures, as well as the predicted leading coefficient formula, up to a $p$-adic unit.

Our conjectures are very closely related to conjectures of Birch and Swinnerton-Dyer type formulated by Bertolini and Darmon in 1996 for Heegner distributions, and as application of our results we also obtain the proof of an inequality in the rank part of their conjectures.

Classification : 11G05, 11R23, 11G16
Mots clés : Elliptic curves, Birch and Swinnerton-Dyer conjecture, Heegner points, $p$-adic $L$-functions

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