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Modularity of the Consani-Scholten quintic. With an appendix by José Burgos Gil and Ariel Pacetti. (English) Zbl 1337.11029

Summary: We prove that the Consani-Scholten quintic, a Calabi-Yau threefold over \(\mathbb Q\), is Hilbert modular. For this, we refine several techniques known from the context of modular forms. Most notably, we extend the Faltings-Serre-Livné method to induced four-dimensional Galois representations over \(\mathbb Q\). We also need a Sturm bound for Hilbert modular forms; this is developed in an appendix by José Burgos Gil and the second author.

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
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14J32 Calabi-Yau manifolds (algebro-geometric aspects)

Citations:

Zbl 1115.14031

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