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.
Publié le : 2005-03-14
Classification:
Group schemes,
Witt vectors,
lattices,
Hilbert class field,
20G25,
20G99,
14L15
@article{1113234835,
author = {Haboush, William J.},
title = {Infinite dimensional algebraic geometry: algebraic structures on {$p$}-adic groups and their homogeneous spaces},
journal = {Tohoku Math. J. (2)},
volume = {57},
number = {1},
year = {2005},
pages = { 65-117},
language = {en},
url = {http://dml.mathdoc.fr/item/1113234835}
}
Haboush, William J. Infinite dimensional algebraic geometry: algebraic structures on {$p$}-adic groups and their homogeneous spaces. Tohoku Math. J. (2), Tome 57 (2005) no. 1, pp. 65-117. http://gdmltest.u-ga.fr/item/1113234835/