Number Theory Seminar

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.