# Current

The seminar is held on Tuesdays from 11 to 12 and is organised by Razvan Barbulescu and Wessel van Woerden. Unless otherwise stated it takes place in room 2 of IMB. To get announcements, you can subscribe to the mailing list of the Bordeaux number theory seminars. Last minute changes may appear first on the IMB website.

- 2024-10-0811:00Salle 2Sabrina Kunzweiler (CANARI)Computing modular polynomials by deformationThe classical modular polynomial $\Phi_N$ parametrizes pairs of elliptic curves connected by an isogeny of degree $N$. They play an important role in algorithmic number theory, and are used in many applications, for example in the SEA point counting algorithm. This talk is about a new method for computing modular polynomials. It has the same asymptotic time complexity as the currently best known algorithms, but does not rely on any heuristics. The main ideas of our algorithm are: the embedding of $N$-isogenies in smooth-degree isogenies in higher dimension, and the computation of deformations of isogenies. The talk is based on a joint work with Damien Robert.
- 2024-10-1511:00Salle 2Damien Robert (CANARI)The module action on abelian varietiesIn a category enriched in a closed symmetric monoidal category, the power object construction, if it is representable, gives a contravariant monoidal action. We first survey the construction, due to Serre, of the power object by (projective) Hermitian modules on abelian varieties. The resulting action, when applied to a primitively oriented elliptic curve, gives a contravariant equivalence of categories (Jordan, Keeton, Poonen, Rains, Shepherd-Barron and Tate). We then give several applications of this module action: 1) We first explain how it allows to describe purely algebraically the ideal class group action on an elliptic curve or the Shimura class group action on a CM abelian variety over a finite field, without lifting to characteristic 0. 2) We then extend the usual algorithms for the ideal action to the case of modules, and use it to explore isogeny graphs of powers of an elliptic curve in dimension up to 4. This allows us to find new examples of curves with many points. (This is a joint work with Kirschmer, Narbonne and Ritzenthaler). 3) Finally, we give new applications for isogeny based cryptography. We explain how, via the Weil restriction, the supersingular isogeny path problem can be recast as a rank 2 module action inversion problem. We also propose ⊗-MIKE a novel NIKE (non interactive isogeny key exchange) that only needs to send j-invariants of supersingular curves, and compute a dimension 4 abelian variety as the shared secret.
- 2024-11-0511:00Salle 2Arthur Herlédan Le Merdy (ÉNS Lyon)TBA
- 2024-11-1211:00Salle 2Sam Frengley (University of Bristol)TBA