## 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].

### MSC:

 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

### Keywords:

Shimura varieties; Galois representations; theta lifts; L-series
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