Modularity of fibres in rigid local systems

Henri Darmon

Review from Zentralblatt MATH:

The paper under review considers two-dimensional $\ell$-adic representations of the absolute Galois group $G_K$ of a totally real number field $K$ occurring in `rigid families' of such representations. This means that these representations are obtained by specialising a representation $\rho$ of $G_{K(t)}$ at points $x \in {\Bbb P}^1(K)$, where it is unramified, which is assumed to be the case for all $x \notin \{0, 1, \infty\}$. Under some technical conditions on the monodromy at the three exceptional points, the author proves that the representations $\rho[x]$ are modular for all $x \in {\Bbb P}^1({\Bbb Q}) \setminus \{0, 1, \infty\}$, provided certain instances of the `lifting conjecture' are true. This conjecture asserts that an $\ell$-adic representation $\rho$ is modular if its residual representation $\bar{\rho}$ is modular, and has been proved in some cases, see for example {\it C. M. Skinner} and {\it A. Wiles} [Proc. Natl. Acad. Sci. USA 94, 10520-10527 (1997; Zbl 0924.11044)]. The main idea of the proof is to identify the representations under consideration with representations on Tate modules of certain abelian varieties.

Reviewed by Michael Stoll

Keywords: modularity of Galois representations over totally real fields

Classification (MSC2000): 11F80 11F41 11G10 11F33

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