Four dimensional Galois representations. (English) Zbl 1097.11027

Tilouine, Jacques (ed.) et al., Automorphic forms (II). The case of the group \(\text{GSp}(4)\). Paris: Société Mathématique de France (SMF) (ISBN 2-85629-184-8/pbk). Astérisque 302, 67-150 (2005).
If \(\mathbb A\) denotes the ring of rational adeles, it is well-known that two-dimensional \(\lambda\)-adic representations are associated to irreducible cuspidal automorphic representations of the group \(\text{GL} (2, \mathbb A)\) whose archimedian component is a discrete series representation. In this case the condition at the archimedian place leads to the study of classical holomorphic cusp forms of weight \(k \geq 2\). In this paper the author studies similar results for the group \(\text{GSp} (4)\) of symplectic similitudes. In particular, he constructs four-dimensional irreducible mixed \(\ell\)-adic representations of the absolute Galois group of \(\mathbb Q\), which are attached to irreducible cuspidal automorphic representations \(\Pi\) of \(\text{GSp} (4)\) with \(\Pi_\infty\) belonging to the discrete series. He then discusses some of the properties of such \(\ell\)-adic representations.
For the entire collection see [Zbl 1089.11003].


11F80 Galois representations
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11G18 Arithmetic aspects of modular and Shimura varieties
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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