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Adjoint motives of modular forms and the Tamagawa number conjecture · arXivDesk

2512.02348Dec 2, 2025109 pages. This is the longer version of the paper: The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 5, 663-727. This version contains details in the foundational part (Section 1) that were not included in the published version

Adjoint motives of modular forms and the Tamagawa number conjecture

Fred Diamond, Matthias Flach, Li Guo

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 $M$

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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

λ

-part of the Tamagawa number conjecture of Bloch and Kato for

L(A,0)

and

L(A,1)

. Here

λ

is any prime of

K

not dividing

Nk!

, and so that the mod

λ

representation associated to

f

is absolutely irreducible when restricted to the Galois group over

\mathbb{Q}(\sqrt{(-1)^{(\ell-1)/2}\ell})

where

λ\mid \ell

. The method also establishes modularity of all lifts of the mod

λ

representation which are crystalline of Hodge-Tate type

(0,k-1)

.

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