Thursday, May 17, 2018
Number Theory Seminar
A split-complex analogue of abelian varieties
Zavosh Amir-Khosravi, Department of Mathematics, Caltech
The split-complex numbers are the split quadratic extension of the reals. We define the analogue of a complex abelian variety over this ring as a polarized split-complex torus, and show that a number of classical results about abelian varieties have split-complex analogues. For instance, principally polarized split-complex tori are parametrized by an arithmetic quotient of the symmetric space of $O(n,n)$, and there is a natural notion of real multiplication, corresponding to special points on the moduli space. We describe a construction relating these objects to abelian varieties, and discuss reciprocity laws, $p$-divisible groups, and a connection with abelian varieties over R.