## The meromorphic continuation of the zeta function of Siegel modular threefolds over totally real fields.(English)Zbl 1320.11109

From the introduction: Let $$S_K:=S_{G,K}$$ be the Siegel modular threefolds associated to $$G:=\mathrm{Gsp}_4$$ and to some open compact subgroup $$K$$ of $$G(\mathbb A_{\mathbb Q,f})$$, where $$\mathbb A_{\mathbb Q,f}$$ is the finite part of the ring of adeles $$\mathbb A_{\mathbb Q}$$ of $$\mathbb Q$$.
In this article we prove the meromorphic continuation of the zeta function of $$S_{K/F}$$, where $$F$$ is an arbitrary totally real number field. In order to shown this result we use the potential modularity for some $$l$$-adic representations of the absolute Galois group of totally real number fields [T. Barnet-Lamb et al., Ann. Math. (2) 179, No. 2, 501–609 (2014; Zbl 1310.11060)].

### MSC:

 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F80 Galois representations 11R42 Zeta functions and $$L$$-functions of number fields 11R80 Totally real fields

Zbl 1310.11060
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### References:

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