Kamal Khuri-Makdisi
An analog of the Edwards model for Jacobians of genus 2 curves
The Edwards model of an elliptic curve E is a certain embedding of E into the product P1 x P1 of two projective lines. One can often arrange for the points at infinity on E to not be rational over the ground field k, in which case the set E(k) of rational points lies in the affine plane A1 x A1, and the group law on E can be given by a single formula that works for all of E(k). My talk will discuss these issues for elliptic curves, and will then report on joint work with E. V. Flynn (arxiv:2211.01450) in which we generalize this construction to a general principally polarized abelian surface J. We obtain a model of J as a subvariety of P3 x P3, described by 15 equations of small bidegree in the 8 variables on P3 x P3. This is considerably fewer than the usual 72 quadric equations in 16 variables for an abelian surface embedded in P15.