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)].


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
Full Text: DOI Euclid


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