Section: New Results

Complex multiplication of abelian varieties and elliptic curves

Participants : Razvan Barbulescu, Sorina Ionica, Chloe Martindale, Enea Milio, Damien Robert.

In [16], Sorina Ionica, former postdoc of the team, and Emmanuel Thomé look at the structure of isogeny graphs of genus 2 Jacobians with maximal real multiplication. They generalise a result of Kohel's describing the structure of the endomorphism rings of the isogeny graph of elliptic curves. Their setting considers genus 2 jacobians with complex multiplication, with the assumptions that the real multiplication subring is maximal and has class number 1. Over finite fields, they derive a depth first search algorithm for computing endomorphism rings locally at prime numbers, if the real multiplication is maximal.

Antonin Riffaut examines in [18] whether there are relations defined over $\mathbb{Q}$ that link (additively or multiplicatively) different singular moduli $j\left(\tau \right)$, invariants of elliptic curves with complex multiplication by different quadratic rings.

In [34], Chloe Martindale presents an algorithm to compute higher dimensional Hilbert modular polynomials. She also explains applications of this algorithm to point counting, walking on isogeny graphs, and computing class polynomials.

In [28], Razvan Barbulescu and Sudarshan Shinde (Sorbonne Université) make a complete list of the 1525 infinite families of elliptic curves without CM which have a particular behaviour in the ECM factoring algorithm, the 20 previously known families having been found by ad-hoc methods. The new idea was to use the characterisation of ECM-friendly families in terms of their Galois image and to use the recent progress in the topic of Mazur's program. In particular, for some of the families mentioned theoretical in the literature the article offers the first publication of explicite equations.

E. Milio and D. Robert updated their paper [35] on computing cyclic modular polynomials.