Higher-dimensional modular equations and point counting on abelian surfaces
SEMINARS, NUMBER THEORY
Jean Kieffer - Harvard University
Given a prime number l, the elliptic modular polynomial of level l is an explicit equation for the locus of elliptic curves related by an l-isogeny. These polynomials have a large number of algorithmic applications: in particular, they are an essential ingredient in the celebrated SEA algorithm for counting points on elliptic curves over finite fields of large characteristic.
More generally, modular equations describe the locus of isogenous abelian varieties in certain moduli spaces called PEL Shimura varieties. We will present upper bounds on the size of modular equations in terms of their level, and outline a general strategy to compute an isogeny A->A' given a point (A,A') where the modular equations are satisfied. This generalizes well-known properties of elliptic modular polynomials to higher dimensions.
The isogeny algorithm is made fully explicit for Jacobians of genus 2 curves. In combination with efficient evaluation methods for modular equations in genus 2 via complex approximations, this gives rise to point counting algorithms for (Jacobians of) genus 2 curves whose
asymptotic costs, under standard heuristics, improve on previous results.