On the order of vanishing of newforms at cusps

Abstract

Let E be an elliptic curve over Q of conductor N. We obtain an explicit formula, as a product of local terms, for the ramification index at each cusp of a modular parametrization of E by X0(N). Our formula shows that the ramification index always divides 24, a fact that had been previously conjectured by Brunault as a result of numerical computations. In fact, we prove a more general result which gives the order of vanishing at each cusp of a holomorphic newform of arbitary level, weight and character, provided that its field of rationality satisfies a certain condition. The above result relies on a purely p-adic computation of possibly independent interest. Let F be a non-archimedean local field of characteristic 0 and π an irreducible, admissible, generic representation of GL2(F). We introduce a new integral invariant, which we call the vanishing index and denote eπ(l), that measures the degree of “extra vanishing” at matrices of level l of the Whittaker function associated to the new-vector of π. Our main local result writes down the value of eπ(l) in every case.