Plus/minus p-adic L-functions for Hilbert modular forms

Abstract

R. Pollack constructed in Pollack (2003) [13] plus/minus p-adic L-functions for elliptic modular forms, which are p-adically bounded, when the Hecke eigenvalues at p are zero (the most super-singular case). The goal of this work is to generalize his construction to Hilbert modular forms. We find a suitable condition for Hilbert modular forms corresponding to the vanishing of p-th Hecke eigenvalue in elliptic modular form case, which guarantees the existence of plus/minus p-adic L-functions which are p-adically bounded. As an application, we construct cyclotomic plus/minus p-adic L-functions for modular elliptic curves over a totally real field F under the assumption that a(p)(E) = 0 for each prime p dividing p. We formulate a cyclotomic plus/minus Iwasawa main conjecture for such elliptic curves. (C) 2011 Elsevier Inc. All rights reserved.