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$p$-adic $L$-functions for $\text{GL}_{2}$

Part of: Zeta and $L$-functions: analytic theory Discontinuous groups and automorphic forms Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  07 January 2019

Daniel Barrera Salazar  and
Chris Williams
Daniel Barrera Salazar
Affiliation:
Universitat Politécnica de Catalunya, Campus Nord, Calle Jordi Girona, 1-3, 08034 Barcelona, Spain Email: daniel.barrera.salazar@upc.edu
Chris Williams
Affiliation:
Mathematics Department, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK Email: christopher.williams@imperial.ac.uk
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Abstract

Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct $p$-adic $L$-functions for non-critical slope rational modular forms, the theory has been extended to construct $p$-adic $L$-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors, respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, which moreover interpolates critical values of the $L$-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the $p$-adic $L$-function of the eigenform to be this distribution.

Keywords

automorphic formGL(2)p-adic L-functionL-functionmodular symboloverconvergentcohomologyautomorphic cyclecontrol theoremL-valuedistribution

MSC classification

Primary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F85: $p$-adic theory, local fields 11S40: Zeta functions and $L$-functions 11M41: Other Dirichlet series and zeta functions
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 5 , October 2019 , pp. 1019 - 1059
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

Author D. B. S. was funded by the Centre de Recherches Mathématiques in Montreal and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 682152). Author C. W. was supported by an EPSRC DTG doctoral grant at the University of Warwick.

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