Supersingular abelian varieties and modular forms

Опубликовано 7 сентября 2016, 17:10

We give a survey on old and new results on relations between geometric invariants of principally polarized supersingular abelian varieties and arithmetic invariants of quaternion hermitian forms (such as the numbers of polarizations and irreducible components of the supersingular locus, the field of definition, existence of curves with many rational points, class numbers, type numbers, and Hecke operators). Conjecturally quaternion hermitian forms are related with Siegel modular forms by Langlands philosophy. So we also see where Siegel modular forms appear in the above theory.