On a conjecture of Darmon-Rotger in the adjoint CM case

Abstract: Let $E$ be an elliptic curve over $\mathbf{Q}$, and suppose that $L(E,s)$ has sign $+1$ in its functional equation and vanishes at $s=1$. Let $p>3$ be a prime of good ordinary reduction for $E$. A construction of Darmon-Rotger attaches to $E$ and an auxiliary weight one cuspidal eigenform $g$ a Selmer class $\kappa_p(E,g,g^*)\in\mathrm{Sel}(\mathbf{Q},V_pE)$. Assuming that $L(E,{\rm ad}^0(g),1)\neq 0$, they conjectured that the following are equivalent: (1) $\kappa_p(E,g,g^*)\neq 0$, (2) ${\rm dim}_{\mathbf{Q}_p}\mathrm{Sel}(\mathbf{Q},V_pE)=2$.

In this talk I will outline a proof of Darmon-Rotger's conjecture when $g$ has CM and the Tate-Shafarevich group of $E$ has finite $p$-primary part (and some mild additional hypotheses). If time permits, I'll also talk about the extension of these results to the case of supersingular primes $p$. Based on joint work with Ming-Lun Hsieh.

Spanishnumber theory

Audience: researchers in the topic

Coloquio Latinoamericano de Teoría de Números

Series comments: El objetivo de este coloquio es fomentar el desarrollo de la teoría de números en latinoamérica, y sus colaboraciones, por medio de exposiciones de trabajos de investigación a cargo de personas pertenecientes a distintos centros de investigación, con intereses comunes en teoría de números y áreas afines.

La presentación estará seguida por un "café virtual" al que están invitados todos los participantes.