Christophe Ritzenthaler (Universitat Autonoma de Barcelone), "Where are the Jacobians ?"

Joint work with E. Howe and Enric Nart.

An isogeny class A of abelian surfaces over a finite field F_q is determined by a Weil polynomial of the form x^4+ax^3+bx^2+aqx+q^2. For which values of (a,b) there is a projective smooth genus 2 curve over F_q whose Jacobian lies in A? We give a complete solution to this problem. During the talk, we focus on the following particular cases :

- For simple supersingular surfaces that split over F_q^2 we use results of Shimura, Oort and Ibukiyama-Katsura-Oort on supersingular abelian surfaces and their polarizations. We descend principally polarized surfaces from \bar{F_q} to F_q and we control the isogeny class of the descended surface; to this end we use results of Hashimoto and Ibukiyama on mass formulas for quaternion hermitian forms with a given group of automorphisms.

- For the split case one uses a result of Kani characterizing when two elliptic curves can be tied along respective torsion subgroups to get a common covering by a curve of genus two. In some cases we need to study twists of Dieudonn\'e modules of supersingular elliptic curves to check that certain curves belonging to different isogeny classes cannot be tied along their p-torsion subgroups.