Adjoint motives of modular forms and the Tamagawa number conjecture

TL;DR

This paper verifies a key part of the Tamagawa number conjecture for the adjoint motive of modular forms using Taylor-Wiles methods, establishing new cases of the conjecture and modularity results.

Contribution

It extends the verification of the Tamagawa number conjecture to the $ ext{lambda}$-part for adjoint motives of modular forms and proves modularity of certain crystalline lifts.

Findings

Verified the $ ext{lambda}$-part of the Tamagawa number conjecture for $L(A,0)$ and $L(A,1)$.

Established modularity of all crystalline lifts of the mod $ ext{lambda}$ representation with specified Hodge-Tate type.

Constructed integral structures in the realizations of the motives involved.

Abstract

Let $f$ be a newform of weight $k\geq 2$, level $N$ with coefficients in a number field $K$, and $A$ the adjoint motive of the motive $M$ associated to $f$. We carefully discuss the construction of the realisations of $M$ and $A$, as well as natural integral structures in these realisations. We then use the method of Taylor and Wiles to verify the $\lambda$-part of the Tamagawa number conjecture of Bloch and Kato for $L(A,0)$ and $L(A,1)$. Here $\lambda$ is any prime of $K$ not dividing $Nk!$, and so that the mod $\lambda$ representation associated to $f$ is absolutely irreducible when restricted to the Galois group over $\mathbb{Q}(\sqrt{(-1)^{(\ell-1)/2}\ell})$ where $\lambda\mid \ell$. The method also establishes modularity of all lifts of the mod $\lambda$ representation which are crystalline of Hodge-Tate type $(0,k-1)$.