Infinite dimensional algebraic geometry: algebraic structures on {$p$}-adic groups and their homogeneous spaces

Abstract

Let $k$ denote the algebraic closure of the finite field, $\mathbb F_p,$ let $\mathcal O$ denote the Witt vectors of $k$ and let $K$ denote the fraction field of this ring. In the first part of this paper we construct an algebraic theory of ind-schemes that allows us to represent finite $K$ schemes as infinite dimensional $k$-schemes and we apply this to semisimple groups. In the second part we construct spaces of lattices of fixed discriminant in the vector space $K^n.$ We determine the structure of these schemes. We devote particular attention to lattices of fixed discriminant in the lattice, $p^{-r}\mathcal O^n,$ computing the Zariski tangent space to a lattice in this scheme and determining the singular points.