We determine the group structure of the normalizer of Γ0(N) in SL2(R) modulo Γ0(N). These results correct the Atkin-Lehner statement [1, Theorem 8].

In contrast to this situation, there is currently no satisfactory theory of local newforms for the group GSp(2, F ). As a consequence, there is no analogue of Atkin–Lehner theory for Siegel modular forms of degree 2. In this paper we shall present such a theory for the “square-free” case. In the local context this means that the representations in question are assumed to have nontrivial Iwahori...

We prove that the non-CM Q-abelian surfaces whose endomorphism algebra is a quaternion algebra are parametrized, up to isogeny, by the rational points of the quotient of certain Shimura curves by the group of their Atkin-Lehner involutions.

We first obtain the dimension formulas for spaces of holomorphic modular forms with character Fricke group $\Gamma _0^+(N)$, and then _0^*(N)$ all Atkin–Lehner involutions added in a particular case.

Our goal is to give an introduction to modular forms and to discuss the role of Hecke operators in the theory. We will focus on the parts of the theory that we will need in studying [GZ86]. For more details, one can refer to [DI95]. Recall the following topics from Jeff’s talks: • The Atkin-Lehner involution Wk is an involution on the modular curve Y0(N) = h/Γ0(N) and permute the cusps. • Eisen...

Using Serre’s proposed complement to Shih’s Theorem, we obtain PSL2(Fp) as a Galois group over Q for at least 614 new primes p. Assuming that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for 3 8 of the primes that were not covered by previous results; it would also suffice to assume a certain (plausible, and perhaps tractable) conjecture conc...

Let ` be one of the ten integers such that the sum of their divisors divide 24. For each such `, (except 15) we give a map from an algebra of polynomial invariants of some finite group to the algebra of modular forms invariant under the Atkin–Lehner group of level `. These maps are motivated and inspired by constructions of modular lattices from self-dual codes over rings. This work generalizes...

If f(z) is a weight k ∈ 1 2 Z meromorphic modular form on Γ0(N) satisfying f(z) = ∑