On the Construction of a Finite Siegel Space

Abstract

We construct a finite analogue of classical Siegel's Space. This is made by generalizing Poincare half plane construction for a quadratic field extension E superset of F, considering in this case an involutive ring A, extension of the ring fixed points A(0) = A(Gamma), (Gamma an order two group of automorphisms of A), and the generalized special linear group SL*(2, A), which acts on a *- plane P-A. Classical Lagrangians for finite dimensional spaces over a finite field are related with Lagrangians for PA. We show SL* (2, A) acts transitively on PA when A is a *- euclidean ring, and we study extensibly the case where A = M-n(E). The structure of the orbits of the action of the symplectic group over F on Lagrangians over a finite dimensional space over E are studied.