On the Tamagawa number conjecture for newforms at Eisenstein primes

TL;DR

This paper proves a rank 0 Tamagawa number formula for higher weight modular forms at Eisenstein primes, extending previous results and discussing partial progress towards rank 1 cases under certain hypotheses.

Contribution

It extends the Tamagawa number conjecture to higher weight modular forms at Eisenstein primes and explores partial results for rank 1 cases, including a conditional higher weight p-converse theorem.

Findings

Proved a rank 0 Tamagawa number formula for modular forms at Eisenstein primes.

Discussed partial results towards a rank 1 formula under standard hypotheses.

Derived a conditional higher weight p-converse theorem from Iwasawa theory.

Abstract

We extend the results of [CGLS22] to higher weight modular forms and prove a rank $0$ Tamagawa number formula (also known as the Bloch-Kato conjecture) for modular forms at good Eisenstein primes, under some technical assumption on periods. Under standard hypotheses (i.e. the injectivity of the $p$-adic Abel-Jabobi map and the non-degeneracy of the Gillet-Soul\'e height pairing), we also discuss some partial results towards a rank $1$ result. A conditional higher weight $p$-converse theorem to Gross-Zagier-Zhang-Kolyvagin-Nekov\'a\v{r} is also obtained as a consequence of the anticyclotomic Iwasawa Main Conjectures.