$P$-adic $L$-functions of Bianchi modular forms

Abstract

The theory of overconvergent modular symbols, developed by Rob Pollack and Glenn Stevens, gives a beautiful and effective construction of the $p$-adic $L$-function of a modular form. In this paper, we give an analogue of their results for Bianchi modular forms, that is, modular forms over imaginary quadratic fields. In particular, we prove control theorems that say that the canonical specialisation map from overconvergent to classical Bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable Hecke operators. We also give an explicit link between the classical modular symbol attached to a Bianchi modular form and critical values of its $L$-function, which then allows us to construct $p$-adic $L$-functions of Bianchi modular forms.

41 pages. I have relaxed a condition, previously assumed throughout, that the primes above p are principal. I have also made various minor improvements and corrections throughout

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• P-adic Asai L-functions of Bianchi modular forms
• Functional equation of the $p$-adic $L$-function of Bianchi modular forms
• Stark-Heegner cycles attached to Bianchi modular forms
• Non-cuspidal Bianchi modular forms and Katz $p$-adic $L$-functions
• Bianchi modular symbols and $p$-adic $L$-functions
• Bianchi period polynomials: Hecke action and congruences
• On $p$-adic $L$-functions for symplectic representations of GL(N) over number fields