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On special zeros of \(p\)-adic \(L\)-functions of Hilbert modular forms. (English) Zbl 1392.11027
Summary: Let \(E\) be a modular elliptic curve over a totally real number field \(F\). We prove the weak exceptional zero conjecture which links a (higher) derivative of the \(p\)-adic \(L\)-function attached to \(E\) to certain \(p\)-adic periods attached to the corresponding Hilbert modular form at the places above \(p\) where \(E\) has split multiplicative reduction. Under some mild restrictions on \(p\) and the conductor of \(E\) we deduce the exceptional zero conjecture in the strong form (i.e. where the automorphic \(p\)-adic periods are replaced by the \(\mathcal{L}\)-invariants of \(E\) defined in terms of Tate periods) from a special case proved earlier by C. P. Mok [Compos. Math. 145, No. 1, 1–55 (2009; Zbl 1247.11071)]. Crucial for our method is a new construction of the \(p\)-adic \(L\)-function of \(E\) in terms of local data.

MSC:
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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