2nd May 2006
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.