On the construction of a finite Siegel space

Abstract

We construct a finite analogue of classical Siegel’s Space. This is made by generalizing Poincar´e half plane construction for a quadratic field extension E ⊃ F, considering in this case an involutive ring A, extension of the ring fixed points A0 = AΓ, (Γ an order two group of automorphisms of A), and the generalized special linear group SL∗(2, A), which acts on a certain ∗− plane PA. 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 = Mn(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.