On the order of vanishing of newforms at cusps

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 arbitary level, weight and character, provided 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 and $\pi$ an irreducible, admissible, generic representation of ${\mathrm{GL}}_2(F)$. We introduce a new integral invariant, which we call the \emph{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 newvector of $\pi$. Our main local result writes down the value of $e_{\pi }(l)$ in every case.

Added Remark 2.20, which clarifies potential priority issues regarding a particular formula

References in corpus (2)

• Notes on the arithmetic of Hilbert modular forms
• Hybrid sup-norm bounds for Maass newforms of powerful level

Cited by in corpus (8)

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• On the size of p-adic Whittaker functions
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• Fourier expansions at cusps
• Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity
• Motivic action on coherent cohomology of Hilbert modular varieties
• An explicit conductor formula for ${\mathrm{GL}}_n\times {\mathrm{GL}}_1$