A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a 2-isogeny of elliptic curves.

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.

In 2000 T. Satoh gave the first p–adic point counting algorithm for elliptic curves over finite fields. Satoh’s algorithm was followed by the SST algorithm and furthermore by the AGM and MSST algorithms for characteristic two only. All four algorithms are important to Elliptic Curve Cryptography. In this paper we present a general framework for p–adic point counting and we apply it to elliptic ...

Abstract Given an elliptic curve E and a finite subgroup G, Vélu’s formulae concern to a separable isogeny IG : E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P +G as the difference between the abscissa of IG(P ) and the first elementary symmetric polynomial on the abscissas of th...

Given a rational point on a curve in a rational isogeny class, a natural question concerns the field of definition of its pre-images. The multiplication by m endomorphism is a powerful and much-used tool. The pre-images for this map are found by factorizing a monic polynomial of degree m. For m = 2, Everest and King gave examples where the existence of a quadratic factor coincided with the exis...

For a complex abelian surface A with endomorphism ring isomorphic to the maximal order in a quartic CM-field K, the Igusa invariants j1(A), j2(A), j3(A) generate an unramified abelian extension of the reflex field of K. In this paper we give an explicit geometric description of the Galois action of the class group of this reflex field on j1(A), j2(A), j3(A). Our description can be expressed by ...

Presence of quantum computers is a real threat against the security of currently used public key cryptographic algorithms such as RSA and Elliptic curve cryptography. Isogeny computation on elliptic curves is believed to be difficult even on a quantum computer, and hence isogeny-based protocols represent one of the few truly practical approaches to constructing quantum-resistant cryptosystems. ...

Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges are `-isogenies. Algorithms allowing to travel on these graphs were developed by Kohel in his thesis (1996) and later on, by Fouquet and Morain (2001). However, up to now, no method was known, to predict, before taking a step on the volcano, the direction of this step. Hence, in Kohel's and Fouquet-Morain algorithms...