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

Francesc Castella

TL;DR

This work proves the $p$-part of the Birch–Swinnerton-Dyer formula for CM elliptic curves $E/F$ under precise CM and abelian-torsion hypotheses, and extends the framework to CM abelian varieties and CM modular forms. It builds an anticyclotomic Iwasawa Main Conjecture for self-dual pairs $(g,\chi)$ via square-root $p$-adic $L$-functions and Katz two-variable $L$-functions, hinges on generalized Heegner cycles, and leverages explicit reciprocity laws to connect $p$-adic invariants with central $L$-values. The approach yields a Gross–Zagier–Kolyvagin-type p-converse in this CM setting and provides a pathway to the equivariant Tamagawa number conjecture in analytic rank 1 for CM motives. By specializing to weight 2, it derives the $p$-part BSD for CM abelian varieties, supported by period comparisons and Néron–Tate height pairings, with broader implications for higher-weight CM modular forms. Overall, the paper integrates Iwasawa theory, $p$-adic $L$-functions, and Euler systems to establish precise arithmetic–analytic correspondences in the CM context and beyond.

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.

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

TL;DR

This work proves the -part of the Birch–Swinnerton-Dyer formula for CM elliptic curves under precise CM and abelian-torsion hypotheses, and extends the framework to CM abelian varieties and CM modular forms. It builds an anticyclotomic Iwasawa Main Conjecture for self-dual pairs via square-root -adic -functions and Katz two-variable -functions, hinges on generalized Heegner cycles, and leverages explicit reciprocity laws to connect -adic invariants with central -values. The approach yields a Gross–Zagier–Kolyvagin-type p-converse in this CM setting and provides a pathway to the equivariant Tamagawa number conjecture in analytic rank 1 for CM motives. By specializing to weight 2, it derives the -part BSD for CM abelian varieties, supported by period comparisons and Néron–Tate height pairings, with broader implications for higher-weight CM modular forms. Overall, the paper integrates Iwasawa theory, -adic -functions, and Euler systems to establish precise arithmetic–analytic correspondences in the CM context and beyond.

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 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 analogue in this setting of Skinner's -converse to the theorem of Gross--Zagier and Kolyvagin.
Paper Structure (44 sections, 32 theorems, 226 equations)

This paper contains 44 sections, 32 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. About the proofs
  4. About the hypotheses
  5. Acknowledgements
  6. $p$-adic $L$-functions
  7. $p$-adic $L$-functions for self-dual pairs
  8. Self-dual pairs
  9. CM periods
  10. $p$-adic interpolation
  11. Katz $p$-adic $L$-functions
  12. Two-variable $p$-adic $L$-functions
  13. Anticyclotomic projection
  14. Factorization
  15. Selmer groups and Main Conjectures
  16. ...and 29 more sections

Key Result

Theorem A

Let $E$ be an elliptic curve over a number field $F$ with complex multiplication by the ring of integers of an imaginary quadratic field $K$, ${\rm End}_F(E)\simeq\mathcal{O}_K$, such that $F(E_{\rm tors})/K$ is abelian. Let $\psi_E:F^\times\backslash\mathbb{A}_F^\times\rightarrow\mathbb{C}^\times$ and let $p\nmid 6h_K$ be a prime split in $K$, where $h_K:=\#{\rm Pic}(\mathcal{O}_K)$ is the class

Theorems & Definitions (90)

  • Theorem A
  • Remark 1.1.1
  • Theorem B
  • Theorem C
  • Remark 1.2.1
  • Remark 1.2.2
  • Remark 1.2.3
  • Definition 2.1.1
  • Definition 2.1.2
  • Theorem 2.1.3
  • ...and 80 more