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 describe a family of curves C of genus 2 with a maximal isotropic (Z/5) in J [5], where J is the Jacobian variety of C, and develop the theory required to perform descent via (5, 5)isogeny. We apply this to several examples, where it can shown that non-reducible Jacobians have nontrivial 5-part of the Tate-Shafarevich group.

Given an ordinary elliptic curve on Hesse form over a finite field of characteristic three, we give a sequence of elliptic curves which leads to an effective construction of the canonical lift, and obtain an algorithm for computing the number of points. Our methods are based on the study of an explicitly and naturally given 3-isogeny between elliptic curves on Hesse form.

Let A be an isogeny class of abelian surfaces over Fq with Weil polynomial x4+ax3+bx2+aqx+q2. We show that A does not contain a surface that has a principal polarization if and only if a2 − b = q and b < 0 and all prime divisors of b are congruent to 1 modulo 3.

This is an overview and a preview of the theory of mixed motives of level ≤ 1 explaining some results, projects, ideas and indicating a bunch of problems. Dedicated to Jacob Murre Let k be an algebraically closed field of characteristic zero to start with and let S = Spec(k) denote our base scheme. Recall that Murre [46] associates to a smooth n-dimensional projective variety X over S a Chow co...

Let K be a number field, Galois over Q. A Q-curve over K is an elliptic curve over K which is isogenous to all its Galois conjugates. The current interest in Q-curves, it is fair to say, began with Ribet’s observation [27] that an elliptic curve over Q̄ admitting a dominant morphism from X1(N) must be a Q-curve. It is then natural to conjecture that, in fact, all Q-curves are covered by modular ...

We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree ` (` different from the characteristic) in time quasi-linear with respect to `. This is based in particular on fast algorithms for power series expansion of the Weierstrass ℘-function and related fu...

We show that there are no non-zero semi-stable abelian varieties over Q( √ 5) with good reduction outside 3 and we show that the only semi-stable abelian varieties over Q with good reduction outside 15 are, up to isogeny over Q, powers of the Jacobian of the modular curve X0(15).

The `th modular polynomial, φ`(x, y), parameterizes pairs of elliptic curves with an isogeny of degree ` between them. Modular polynomials provide the defining equations for modular curves, and are useful in many different aspects of computational number theory and cryptography. For example, computations with modular polynomials have been used to speed elliptic curve point-counting algorithms (...

We show that computation of a sequence of Richelot isogenies from specified supersingular Jacobians of genus-2 curves over Fp can be executed in Fp2 or Fp4 . Based on this, we describe a practical algorithm for computing a Richelot isogeny sequence.