dc.contributor.author | Corbett, A | en_US |
dc.contributor.author | Saha, A | en_US |
dc.date.accessioned | 2017-08-14T13:08:24Z | |
dc.date.available | 2017-07-29 | en_US |
dc.date.submitted | 2017-08-05T19:53:24.605Z | |
dc.identifier.issn | 1073-2780 | en_US |
dc.identifier.uri | http://qmro.qmul.ac.uk/xmlui/handle/123456789/25208 | |
dc.description.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. | en_US |
dc.publisher | International Press | en_US |
dc.relation.ispartof | Mathematical Research Letters | en_US |
dc.rights | This is a pre-copyedited, author-produced version of an article accepted for publication in Mathematical Research Letters following peer review. | |
dc.subject | math.NT | en_US |
dc.subject | math.NT | en_US |
dc.title | On the order of vanishing of newforms at cusps | en_US |
dc.type | Article | |
dc.rights.holder | © International Press 2017 | |
pubs.author-url | http://arxiv.org/abs/1609.08939v3 | en_US |
pubs.notes | No embargo | en_US |
pubs.publication-status | Accepted | en_US |
dcterms.dateAccepted | 2017-07-29 | en_US |