Mathematical Research Letters

Volume 25 (2018)

Number 6

On the order of vanishing of newforms at cusps

Pages: 1771 – 1804



Andrew Corbett (Mathematisches Institut, Universität Göttingen, Germany)

Abhishek Saha (School of Mathematical Sciences, Queen Mary University of London, United Kingdom)


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