Tamagawa number conjecture for CM modular forms and Rankin--Selberg convolutions

TL;DR

This paper proves the $p$-part of the Birch--Swinnerton-Dyer formula for CM elliptic curves over number fields in rank 1, extending known results from rational fields to more general CM settings.

Contribution

It extends the proof of the $p$-part of the BSD conjecture to CM elliptic curves over number fields with complex multiplication, for primes split in the CM field, and introduces methods applicable to CM abelian varieties and modular forms.

Findings

Proved the $p$-part of BSD for CM elliptic curves over number fields in rank 1.

Extended results previously known only over $\

Developed new approaches applicable to CM abelian varieties and modular forms.

Abstract

Let $E/F$ be an elliptic curve defined over a number field $F$ with complex multiplication by the ring of integers of an imaginary quadratic field $K$ such that the torsion points of $E$ generate over $F$ an abelian extension of $K$. In this paper we prove the $p$-part of the Birch--Swinnerton-Dyer formula for $E/F$ in analytic rank $1$ for primes $p>3$ split in $K$. This was previously known for $F=\mathbb{Q}$ by work of Rubin as a consequence of his proof of Mazur's Main Conjecture for rational CM elliptic curves, but the problem for $[F:\mathbb{Q}]>1$ remained wide open. The approach introduced in this paper also yields a proof of similar results for CM abelian varieties $A/K$ and for CM modular forms, as well as an analogue in this setting of Skinner's $p$-converse to the theorem of Gross--Zagier and Kolyvagin.