The refined Tamagawa number conjectures for $\mathrm{GL}_2$

Abstract

Let $f$ be a cuspidal newform and $p \geq 3$ a prime such that the associated $p$-adic Galois representation has large image. We establish a new and refined "Birch and Swinnerton-Dyer type" formula for Bloch-Kato Selmer groups of the central critical twist of $f$ via Kolyvagin derivatives of $L$-values instead of complex analytic or $p$-adic variation of $L$-values only under the Iwasawa main conjecture localized at the augmentation ideal. Our formula determines the exact rank and module structure of the Selmer groups and is insensitive to weight, the local behavior of $f$ at $p$, and analytic rank. As consequences, we prove the non-vanishing of Kato's Kolyvagin system and complete a "discrete" analogue of the Beilinson-Bloch-Kato conjecture for modular forms at ordinary primes. We also obtain the higher weight analogue of the $p$-converse to the theorem of Gross-Zagier and Kolyvagin, the $p$-parity conjecture, and a new computational upper bound of Selmer ranks. We also discuss how to formulate the refined conjecture on the non-vanishing of Kato's Kolyvagin system for modular forms of general weight. In the appendix with Robert Pollack, we compute several numerical examples on the structure of Selmer groups of elliptic curves and modular forms of higher weight. Sometimes our computation provides a deeper understanding of Selmer groups than what is predicted by Birch and Swinnerton-Dyer conjecture.