## Section: New Results

### Modular Polynomials of Hilbert Surfaces

Participant : Enea Milio.

InĀ [23], together with Damien Robert from the LFANT team, we describe an evaluation/interpolation approach to compute modular polynomials on a Hilbert surface, which parametrizes abelian surfaces with maximal real multiplication. Under some heuristics we obtain a quasi-linear algorithm. The corresponding modular polynomials are much smaller than the ones on the Siegel threefold. We explain how to compute even smaller polynomials by using pullbacks of Theta functions to the Hilbert surface, and give an application to the CRT method to construct class polynomials.