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

2024/07/16 by Castella, Francesc · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2407.11891

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=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: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.

Cited by

• On the Tamagawa number conjecture for modular forms twisted by anticyclotomic Hecke characters

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