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# $P$-adic $L$-functions for GL$_2$

Published:2018-04-30

• Daniel Barrera Salazar,
Universitat Politécnica de Catalunya, Campus Nord, Calle Jordi Girona, 1-3, 08034 Barcelona, Spain
• Chris Williams,
Mathematics Department, Imperial College London , South Kensington Campus, London, SW7 2AZ, 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, that 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 form, GL(2), p-adic L-function, L-function, modular symbol, overconvergent, cohomology, automorphic cycle, control theorem, L-value, distribution
 MSC Classifications: 11F41 - Automorphic forms on ${\rm GL}(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] 11F67 - Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F85 - $p$-adic theory, local fields [See also 14G20, 22E50] 11S40 - Zeta functions and $L$-functions [See also 11M41, 19F27] 11M41 - Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

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