On the order of vanishing of newforms at cusps
Corbett, A; Saha, A
Date: 20 November 2018
Journal
Mathematical Research Letters
Publisher
International Press
Publisher DOI
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 ...
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.
Computer Science
Faculty of Environment, Science and Economy
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