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Mathematical Research Letters
Volume 25 (2018)
Number 6
On the order of vanishing of newforms at cusps
Pages: 1771 – 1804
DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n6.a4
Authors
Abstract
Let $E$ be an elliptic curve over $\mathbb{Q}$ 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 $X_0 (N)$. 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 arbitrary 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 $\pi$ an irreducible, admissible, generic representation of $GL_2 (F)$. We introduce a new integral invariant, which we call the vanishing index and denote $e_{\pi} (l)$, that measures the degree of “extra vanishing” at matrices of level l of the Whittaker function associated to the new-vector of $\pi$. Our main local result writes down the value of $e_{\pi} (l)$ in every case.
Received 19 October 2016
Accepted 4 June 2017
Published 25 March 2019