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Modularity of fibres in rigid local systems. (English) Zbl 0963.11029

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 \(x \in {\mathbb P}^1(K)\), 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 \(x \in {\mathbb P}^1({\mathbb Q}) \setminus \{0, 1, \infty\}\), 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 C. M. Skinner and 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.

MSC:

11F80 Galois representations
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11G10 Abelian varieties of dimension \(> 1\)
11F33 Congruences for modular and \(p\)-adic modular forms

Citations:

Zbl 0924.11044
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