We wish to give a new proof of one of the main results of Atkin-Lehner [1]. That paper depends, among other things, on a slightly strengthened version of Theorem 1 below, which characterizes forms in Sk(Γ0(N)) whose Fourier coefficients satisfy a certain vanishing condition. Our proof involves rephrasing this vanishing condition in terms of representation theory; this, together with an elementa...

Given an algebraic curve C/Q having points everywhere locally and endowed with a suitable involution, we show that there exists a positive density family of prime quadratic twists of C violating the Hasse principle. The result applies in particular to wN -Atkin-Lehner twists of most modular curves X0(N) and to wp-Atkin-Lehner twists of certain Shimura curves XD+.

Abstract We settle a part of the conjecture by Bandini and Valentino [‘On structure slopes Drinfeld cusp forms’, Exp. Math. 31 (2) (2022), 637–651] for $S_{k,l}(\Gamma _0(T))$ when $\mathrm {dim}\ S_{k,l}(\mathrm {GL}_2(A))\leq 2$ . frame check primes $\mathfrak {p}$ higher levels {p}\mathfrak {m}$ , show that level {p} \mathfrak does not hold if {m}\ne A$ $(k,l)=(2,1)$

Let VD be the Shimura curve over Q attached to the indefinite rational quaternion algebra of discriminant D. In this note we investigate the group of automorphisms of VD and prove that, in many cases, it is the Atkin–Lehner group. Moreover, we determine the family of bielliptic Shimura curves ðover Q and over QÞ and we use it to study the set of rational points on VD over quadratic fields. Fina...

We determine the automorphism group of modular curve $X_0^*(N)$, obtained as quotient $X_0(N)$ by its Atkin-Lehner involutions, for all square-free values $N$.

‎We introduce a higher dimensional Atkin-Lehner theory for‎ ‎Siegel-Parahoric congruence subgroups of $GSp(2g)$‎. ‎Old‎ ‎Siegel forms are induced by geometric correspondences on Siegel‎ ‎moduli spaces which commute with almost all local Hecke algebras‎. ‎We also introduce an algorithm to get equations for moduli spaces of‎ ‎Siegel-Parahoric level structures‎, ‎once we have equations for prime l...