We study the finiteness of low degree points on certain modular curves and their Atkin--Lehner quotients, and, as an application, prove modularity elliptic over all but finitely many totally real fields $5$. On way, we a criterion for rational $5$ curve large genus number field using results Abramovich--Harris Faltings subvarieties Jacobians.

In this paper, we prove Hurwitz-Eichler type formulas for Hurwitz class numbers with each level $ M when the modular curve X_0(M) has genus zero. A key idea is to calculate intersection of correspondences in two different ways. generalization Atkin-Lehner involutions \Gamma_0(M) and its subgroup \Gamma_0^{(M')}(M) introduced multiplicities at cusps.

We present explicit models for Shimura curves X D and Atkin-Lehner quotients X D /ωm of them of genus 2. We show that several equations conjectured by Kurihara are correct and compute for them the kernel of Ribet's isogeny J 0 (D) new → J D between the new part of the Jacobian of the modular curve X 0 (D) and the Jacobian of X D .

Let O be a maximal order in a totally indefinite quaternion algebra over a totally real number field. In this note we study the locus QO of quaternionic multiplication byO in the moduli spaceAg of principally polarized abelian varieties of even dimension g with particular emphasis in the two-dimensional case. We describe QO as a union of Atkin-Lehner quotients of Shimura varieties and we comput...

We exhibit for each global field k an algebraic curve over k which violates the Hasse Principle. We can find such examples among Atkin-Lehner twists of certain elliptic modular curves and Drinfeld modular curves. Our main tool is a refinement of the “Twist Anti-Hasse Principle” (TAHP). We then use TAHP to construct further Hasse Principle violations, e.g. among curves over any number field of a...

Let C be the image of a canonical embedding φ of the Atkin-Lehner quotient X + 0 (N) associated to the Fricke involution w N. Suppose φ is defined over the rationals. In this note we give some collinearity relations among rational points of C, for each X + 0 (N) of genus 3 and the first X + 0 (N) of genus 4, for N prime.

We find defining equations for the Shimura curve of discriminant 15 over Z[1/15]. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy groups of a spectrum of “topological automorphic forms” associated to this curve, as well as one associated to a quotient by an Atkin-Lehner involution.

Let B be the indefinite quaternion algebra over Q of discriminant D. Let O be a maximal order in B. Let XD be the Shimura curve over Q attached to O, whose set of complex points is given by XD(C) = (Ô×\B̂× × P)/B×, where P = C\R. As it is well known, suchXD is equipped with a natural group of involutions called the Atkin-Lehner group W (D), where each involution ωn is indexed by the divisors n |...