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.1 There are exactly φ(L, N/L) cusps of denominator L. The cusp at infinity is the unique one of denominator N .

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Abstract. We prove a precise formula relating the Bessel period of certain automorphic forms on GSp 4 (AF ) to a central L-value. This is a special case of the refined Gan-Gross-Prasad conjecture for the groups (SO5, SO2) as set out by Ichino-Ikeda [12] and Liu [14]. This conjecture is deep and hard to prove in full generality; in this paper we succeed in proving the conjecture for forms lifted, via automorphic induction, from GL2(AE) where E is a quadratic extension of F . The case where E = F × F has been previously dealt with by Liu [14].

We consider the Fourier expansion of a Hecke (resp. Hecke–Maaß) cusp form of general level N at the various cusps of $$\Gamma _{0}(N)\backslash \mathbb {H}$$ Γ 0 ( N ) \ H . We explain how to compute these coefficients via the local theory of p-adic Whittaker functions and establish a classical Voronoï summation formula allowing an arbitrary additive twist. Our discussion has applications to bounding sums of Fourier coefficients and understanding the (generalised) Atkin–Lehner relations.