Related papers: Equations of hyperelliptic Shimura curves

We present explicit models for non-elliptic genus one Shimura curves X_0(D, N) with Gamma_0(N)-level structure arising from an indefinite quaternion algebra of reduced discriminant D, and Atkin-Lehner quotients of them. In addition, we…

We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rather general type of quaternionic or- ders closely related to those introduced by…

Let V_D 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 V_D and prove that, in many cases, it is the Atkin-Lehner group.…

We prove an equidistribution result about Hecke orbits on the Picard group of Shimura curves coming from definite quaternion algebras over function fields. In particular, we show the equidistribution of Hecke orbits of supersingular…

In this work, we extend Howard's construction of compatible families of Heegner points to the setting of towers of Gross curves and Shimura curves over totally real fields. Following the strategy of Longo and Vigni, our approach…

The main goal of this article is to give an explicit rigid analytic uniformization of the maximal toric quotient of the Jacobian of a Shimura curve over the field of rational numbers at a prime dividing exactly the level. This result can be…

Given a rational elliptic curve E, a suitable imaginary quadratic field K and a quaternionic Hecke eigenform g of weight 2 obtained from E by level raising such that the sign in the functional equation for L_K(E,s) (respectively, L_K(g,1))…

We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke…

We give some methods for computing equations for certain Shimura curves, natural maps between them, and special points on them. We then illustrate these methods by working out several examples in varying degrees of detail. For instance, we…

We provide a complete enumeration of all quotients of genus $0, 1$ and $2$ of the Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ by non-trivial subgroups of Atkin--Lehner involutions. For all $1270$ genus $1$ quotients $X$ with $N$ squarefree,…

Given a newform f, we extend Howard's results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families…

A conjecture of Coleman implies that only finitely many quaternion algebras over the rational numbers can be the endomorphism $\mathbf{Q}$-algebras of abelian surfaces over the complex numbers which can be defined over $\mathbf{Q}$. One may…

Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves $X_0(D,N)$. In this paper we compute the $\mathbb{Q}$-rational points on the Atkin-Lehner quotients of these curves using a variety of techniques. We…

In this note we consider certain elliptic curves defined over real quadratic fields isogenous to their Galois conjugate. We give a construction of algebraic points on these curves defined over almost totally real number fields. The main…

We prove a general formula for the $p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above $p$. The formula is in terms of the cyclotomic derivative of a…

We present explicit models for Shimura curves X_D and Atkin-Lehner quotients X_D/w_m 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} -->…

We generalize a result of Ribet and Takahashi on the parametrization of elliptic curves by Shimura curves to the Hilbert modular setting. In particular, we study the behaviour of the parametrization of modular abelian varieties by Shimura…

The formula of the title relates $p$-adic heights of Heegner points and derivatives of $p$-adic $L$-functions. It was originally proved by Perrin-Riou for $p$-ordinary elliptic curves over the rationals, under the assumption that $p$ splits…

ATR points were introduced by Darmon as a conjectural construction of algebraic points on certain elliptic curves for which in general the Heegner point method is not available. So far the only numerical evidence, provided by Darmon--Logan…

This is the third of a series of papers relating intersections of special cycles on the integral model of a Shimura surface to Fourier coefficients of Hilbert modular forms. More precisely, we embed the Shimura curve over Q associated to a…