Adjoint L -Values and Primes of Congruence for Hilbert Modular Forms

2002/07/01 by Eknath Ghate · 4 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Analytic Number Theory Research

paper · pdf · doi:10.1023/a:1016562918902

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 \cal 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 \cal 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/ℚ(ε^k-1 - 1) , where ε is a totally positive unit of F .

Cited by

• Algebraicity of adjoint L-functions for quasi-split groups
• Notes on the arithmetic of Hilbert modular forms
• Noncommutative Iwasawa theory arising from Hecke algebras
• Galois representations modulo p and cohomology of Hilbert modular\n varieties

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