Leading terms of anticyclotomic Stickelberger elements and \(p\)-adic periods. (English) Zbl 1451.11039
Summary: Let \( E\) be a quadratic extension of a totally real number field. We construct Stickelberger elements for Hilbert modular forms of parallel weight 2 in anticyclotomic extensions of \( E\). Extending methods developed by S. Dasgupta and M. Spieß [“The Eisenstein cocycle, partial zeta values and Gross-Stark units”, J. Eur. Math. Soc. (to appear)] from the multiplicative group to an arbitrary one-dimensional torus we bound the order of vanishing of these Stickelberger elements from below and, in the analytic rank zero situation, we give a description of their leading terms via automorphic \( \mathcal {L}\)-invariants. If the field \( E\) is totally imaginary, we use the \( p\)-adic uniformization of Shimura curves to show the equality between automorphic and arithmetic \( \mathcal {L}\)-invariants. This generalizes a result of M. Bertolini and H. Darmon [Invent. Math. 131, No. 3, 453–491 (1998; Zbl 0899.11029); Duke Math. J. 98, No. 2, 305–334 (1999; Zbl 1037.11045)] from the case that the ground field is the field of rationals to arbitrary totally real number fields.
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F75 | Cohomology of arithmetic groups |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
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