相关论文: Computation of ATR Darmon points on non-geometrica…

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…

Let $F/F_0$ be a quadratic extension of totally real number fields, and let $E$ be an elliptic curve over $F$ which is isogenous to its Galois conjugate over $F_0$. A quadratic extension $M/F$ is said to be almost totally complex (ATC) if…

We extend the algorithm of Darmon-Green and Darmon-Pollack for computing p-adic Darmon points on elliptic curves to the case of composite conductor. We also extend the algorithm of Darmon-Logan for computing ATR Darmon points to treat…

Darmon points on p-adic tori and Jacobians of Shimura curves over Q were introduced in previous joint works with Rotger as generalizations of Darmon's Stark-Heegner points. In this article we study the algebraicity over extensions of a real…

Stark-Heegner points, also known as Darmon points, were introduced by H. Darmon as certain local points on rational elliptic curves, conjecturally defined over abelian extensions of real quadratic fields. The rationality conjecture for…

We describe an algorithm that computes explicit models of hyperelliptic Shimura curves attached to an indefnite quaternion algebra over Q and Atkin-Lehner quotients of them. It exploits Cerednik-Drinfeld's non-archimedean uniformisation of…

We use Heegner points to prove the existence of nontorsion rational points on the elliptic curve $y^2 = x^3 + D$ for any rational number $D=a/b$ such that $a$ and $b$ are squarefree integers for which $6$, $a$, and $b$ are pairwise…

We study the homogeneous coordinate rings of real multiplication noncommutative tori as defined by A. Polishchuk. Our aim is to understand how these rings give rise to an arithmetic structure on the noncommutative torus. We start by giving…

We show how to construct Darmon points on elliptic curves over totally real fields.

We present new constructions of complex and p-adic Darmon points on elliptic curves over base fields of arbitrary signature. We conjecture that these points are global and present numerical evidence to support our conjecture.

We investigate multi-point algebraic geometric codes defined from curves related to the generalized Hermitian curve introduced by Alp Bassa, Peter Beelen, Arnaldo Garcia, and Henning Stichtenoth. Our main result is to find a basis of the…

Elliptic curves with a known number of points over a given prime field with n elements are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication…

In this paper, we give an elementary new method for determining the rational points on algebraic curves using torsion packets. We also provide examples of curves for which all rational points can be completely determined by our method.

In previous work, the first author developed an algorithm for the computation of Hilbert modular forms. In this paper, we extend this to all totally real number fields of even degree and nontrivial class group. Using the algorithm over…

We exhibit an algorithm to compute equations of an algebraic curve over a computable characteristic 0 field from the power series expansions of its regular 1-forms at a nonrational point of the curve, extending a 2005 algorithm of Baker,…

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…

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all…

Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a…

Let $E$ be a rational elliptic curve, and $K$ be an imaginary quadratic field. In this article we give a method to construct Heegner points when $E$ has a prime bigger than $3$ of additive reduction ramifying in the field $K$. The ideas…

Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K…