Abstract view
Daniel Barrera Salazar,
Universitat Politécnica de Catalunya, Campus Nord, Calle Jordi Girona, 13, 08034 Barcelona, Spain
Chris Williams,
Mathematics Department, Imperial College London , South Kensington Campus, London, SW7 2AZ, UK
Abstract
Since Rob Pollack and Glenn Stevens used overconvergent
modular symbols to construct $p$adic $L$functions for noncritical
slope rational modular forms, the theory has been extended to
construct $p$adic $L$functions for noncritical 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), padic Lfunction, Lfunction, modular symbol, overconvergent, cohomology, automorphic cycle, control theorem, Lvalue, distribution
automorphic form, GL(2), padic Lfunction, Lfunction, modular symbol, overconvergent, cohomology, automorphic cycle, control theorem, Lvalue, distribution

MSC Classifications: 
11F41, 11F67, 11F85, 11S40, 11M41 show english descriptions
Automorphic forms on ${\rm GL}(2)$; Hilbert and HilbertSiegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] Special values of automorphic $L$series, periods of modular forms, cohomology, modular symbols $p$adic theory, local fields [See also 14G20, 22E50] Zeta functions and $L$functions [See also 11M41, 19F27] Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebrogeometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
11F41  Automorphic forms on ${\rm GL}(2)$; Hilbert and HilbertSiegel 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 algebrogeometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
