We present explicit models for Shimura curves X D and Atkin-Lehner quotients X D /ωm 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 → J D between the new part of the Jacobian of the modular curve X 0 (D) and the Jacobian of X D .

We find the number of elliptic curves with a cyclic isogeny of degree n over various number fields by studying the modular curves X0(n). We show that for n = 14, 15, 20, 21, 49 there exist infinitely many quartic fields K such that #Y0(n)(Q) 6= #Y0(n)(K) < ∞. In the case n = 27 we prove that there are infinitely many sextic fields such that #Y0(n)(Q) 6= #Y0(n)(K) < ∞.

The GHS attack is known to map the discrete logarithm problem(DLP) in the Jacobian of a curve C0 defined over the d degree extension kd of a finite field k to the DLP in the Jacobian of a new curve C over k which is a covering curve of C0, then solve the DLP of curves C/k by variations of index calculus algorithms. It is therefore important to know which curve C0/kd is subjected to the GHS atta...

Fix an ordinary abelian variety defined over a finite field. The ideal class group of its endomorphism ring acts freely on the set of isogenous varieties with same endomorphism ring, by complex multiplication. Any subgroup of the class group, and generating set thereof, induces an isogeny graph on the orbit of the variety for this subgroup. We compute (under the Generalized Riemann Hypothesis) ...

This paper treats what we call ‘weak geometric liftings’ of Galois representations associated to abelian varieties. This notion can be seen as a generalization of the idea of lifting a Galois representation along an isogeny of algebraic groups. The weaker notion only takes into account an isogeny of the derived groups and disregards the centres of the groups in question. The weakly lifted repre...

We construct a relative Chow-Künneth decomposition for a conic bundle over a surface such that the middle projector gives the Prym variety of the associated double covering of the discriminant of the conic bundle. This gives a refinement (up to an isogeny) of Beauville’s theorem on the relation between the intermediate Jacobian of the conic bundle and the Prym variety of the double covering.