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

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Název: Tamagawa number conjecture for CM modular forms and Rankin–Selberg convolutions.
Autoři: Castella, Francesc1 (AUTHOR) castella@ucsb.edu
Zdroj: Proceedings of the London Mathematical Society. Oct2025, Vol. 131 Issue 4, p1-50. 50p.
Témata: *MODULAR forms, *ELLIPTIC curves, *MATHEMATICS theorems, *MATHEMATICAL formulas, *NUMBER theory, *ABELIAN varieties
Abstrakt: Let E/F$E/F$ be an elliptic curve defined over a number field F$F$ with complex multiplication by the ring of integers of an imaginary quadratic field K$K$ such that the torsion points of E$E$ generate over F$F$ an abelian extension of K$K$. In this paper, we prove the p$p$‐part of the Birch–Swinnerton‐Dyer formula for E/F$E/F$ in analytic rank 1 for primes p>3$p>3$ split in K$K$. This was previously known for F=Q$F=\mathbb {Q}$ 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:Q]>1$[F:\mathbb {Q}]>1$ remained wide open. The approach introduced in this paper also yields a proof of similar results for CM abelian varieties A/K$A/K$ and for CM modular forms, as well as an analog in this setting of Skinner's p$p$‐converse to the theorem of Gross–Zagier and Kolyvagin. [ABSTRACT FROM AUTHOR]
Databáze: Academic Search Index
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Abstrakt:Let E/F$E/F$ be an elliptic curve defined over a number field F$F$ with complex multiplication by the ring of integers of an imaginary quadratic field K$K$ such that the torsion points of E$E$ generate over F$F$ an abelian extension of K$K$. In this paper, we prove the p$p$‐part of the Birch–Swinnerton‐Dyer formula for E/F$E/F$ in analytic rank 1 for primes p>3$p>3$ split in K$K$. This was previously known for F=Q$F=\mathbb {Q}$ 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:Q]>1$[F:\mathbb {Q}]>1$ remained wide open. The approach introduced in this paper also yields a proof of similar results for CM abelian varieties A/K$A/K$ and for CM modular forms, as well as an analog in this setting of Skinner's p$p$‐converse to the theorem of Gross–Zagier and Kolyvagin. [ABSTRACT FROM AUTHOR]
ISSN:00246115
DOI:10.1112/plms.70089