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

  • Daniel Barrera Salazar (a1) and Chris Williams (a2)

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.

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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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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
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