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

TL;DR

This work develops a refined Tamagawa-number framework for GL$_2$-motives associated to modular forms, establishing a precise BSD-type formula for Bloch–Kato Selmer groups of the central critical twist under large-image and augmentation-localized Iwasawa main conjecture hypotheses. By introducing Kurihara numbers as discrete derivatives of special $L$-values and linking them to Kato’s zeta elements through a new explicit reciprocity law and a relative Kolyvagin-system argument, the authors obtain an exact Selmer structure independent of weight and local behavior at $p$, while bypassing integral $p$-adic Hodge theory. The paper proves a complete corank and structure description of $\mathrm{Sel}(\mathbb{Q}, W_f^{\dagger})$, with consequences including non-vanishing of Kato’s Kolyvagin system, a discrete Beilinson–Bloch–Kato analogue, a $p$-parity result, and a rank-one $p$-converse for higher weight forms; numerical evidence is provided in an appendix. These results extend BSD/IMC insights to higher ranks and weights, offering both theoretical refinements and practical bounds on Selmer ranks, and they propose conjectural refinements that connect local Tamagawa data with analytic fudge factors in a framework that may require further advances in integral $p$-adic Hodge theory.

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.

Theorems & Definitions (78)

• Theorem 1
• Theorem 1.1: Main Theorem I
• proof
• Theorem 1.2: Main Theorem II
• proof
• Remark 1.4
• Corollary 1.5
• proof
• Conjecture 1.6: Beilinson--Bloch--Kato
• Corollary 1.7