We investigate suborbital graphs for an imprimitive action of the Atkin–Lehner group on a maximal subset of extended rational numbers on which a transitive action is also satisfied. Obtaining edge and some circuit conditions, we examine some combinatorial properties of these graphs.

We study the Čerednik-Drinfeld p-adic uniformization of certain AtkinLehner quotients of Shimura curves over Q. We use it to determine over which local fields they have rational points and divisors of a given degree. Using a criterion of Poonen and Stoll we show that the Shafarevich-Tate group of their jacobians is not of square order for infinitely many cases. In [PSt] Poonen and Stoll have sh...

Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves X0(D,N). In this paper we compute the Q-rational points on Atkin–Lehner quotients of these using a variety techniques. We also determine which rational are CM many curves.

Let X be a Shimura curve of genus at least 2. Exploiting Čerednik-Drinfeld’s description of the special fiber of X and the specialization of its Heegner points, we show that, under certain technical conditions, the group of automorphisms of X corresponds to its group of Atkin-Lehner involutions.

We prove that the abelian K-surfaces whose endomorphism algebra is an indefinite rational quaternion algebra are parametrized, up to isogeny, by the K-rational points of the quotient of certain Shimura curves by the group of their Atkin-Lehner involutions. To cite this article: X. Guitart, S. Molina, C. R. Acad. Sci. Paris, Ser. I

— We give an overview of the theory of Heegner points for elliptic curves, and then describe various new ideas that can be used in the computation of rational points on rank 1 elliptic curves. In particular, we discuss the idea of Cremona (following Silverman) regarding recovery a rational point via knowledge of its height, the idea of Delaunay regarding the use of Atkin-Lehner involutions in t...

Abstract We consider the Fourier expansion of a Hecke (resp. Hecke–Maaß) cusp form general level N at various cusps $$\Gamma _{0}(N)\backslash \mathbb {H}$$ Γ 0 ( N ) \ H </m...

We present explicit models for non-elliptic genus one Shimura curves X0(D, N) with Γ0(N)-level structure arising from an indefinite quaternion algebra of reduced discriminant D, and Atkin-Lehner quotients of them. In addition, we discuss and extend Jordan’s work [10, Ch. III] on points with complex multiplication on Shimura curves.

In this note we consider several maps that occur naturally between modular Shimura varieties, Hilbert-Blumenthal varieties and the moduli spaces of polarized abelian varieties when forgetting certain endomorphism structures. We prove that, up to birational equivalences, these forgetful maps coincide with the natural projection by suitable abelian groups of Atkin-Lehner involutions.

We study three families of Atkin-Lehner quotients of quaternionic Shimura curves: X, X 0 (N), and X D+ 1 (N), which serve as moduli spaces of abelian surfaces with potential quaternionic multiplication (PQM) and level N structure. The arithmetic geometry of these curves is similar to, but even richer than, that of the classical modular curves. Two important differences are the existence of a no...