Abstract
Let f be a primitive Hilbert modular cusp form of arbitrary level and parallel weight k, defined over a totally real number field F. We define a finite set of primes \({\mathcal{S}}\) that depends on the weight and level of f, the field F, and the torsion in the boundary cohomology groups of the Borel–Serre compactification of the underlying Hilbert-Blumenthal variety. We show that, outside \(\mathcal{S}\), any prime that divides the algebraic part of the value at s=1 of the adjoint L-function of f is a congruence prime for f. In special cases we identify the ‘boundary primes’ in terms of expressions of the form \(N_{{F \mathord{\left/ {\vphantom {F {\mathbb{Q}}}} \right. \kern-\nulldelimiterspace} {\mathbb{Q}}}} (\varepsilon ^{k - 1} - 1)\), where ε is a totally positive unit of F.
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Ghate, E. Adjoint L-Values and Primes of Congruence for Hilbert Modular Forms. Compositio Mathematica 132, 243–281 (2002). https://doi.org/10.1023/A:1016562918902
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DOI: https://doi.org/10.1023/A:1016562918902