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October 11, 2025

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

Key Points

The paper proves the p-part of the Birch–Swinnerton–Dyer formula for cm modular forms with analytic rank 1.
Key results extend to CM abelian varieties, addressing previously open problems in the field of number theory.
This approach is rooted in the understanding of elliptic curves with complex multiplication, reinforcing connections between different mathematical theories.
The findings bear significance in understanding the broader implications of the Rankin–Selberg convolutions in analytic number theory.

Abstract

Abstract Let be an elliptic curve defined over a number field with complex multiplication by the ring of integers of an imaginary quadratic field such that the torsion points of generate over an abelian extension of . In this paper, we prove the ‐part of the Birch–Swinnerton‐Dyer formula for in analytic rank 1 for primes split in . This was previously known for by work of Rubin as a consequence of his proof of Mazur's Main Conjecture for rational CM elliptic curves, but the problem for remained wide open. The approach introduced in this paper also yields a proof of similar results for CM abelian varieties and for CM modular forms, as well as an analog in this setting of Skinner's ‐converse to the theorem of Gross–Zagier and Kolyvagin.

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

Key Points

Abstract

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