Symplectic group lattices

Let p be an odd prime. It is known that the symplectic group Sp(2n)(p) has two (algebraically conjugate) irreducible representations of degree (p(n) 1)/2 realized over Q (root epsilon p), where epsilon = (-1)((p-1)/2). We st udy the integral lattices related to these representations for the case p(n ) drop 1 mod 4. (The case p(n) drop 3 mod 4 has been considered in a previo us paper.) We show that the class of invariant lattices contains either uni modular or p-modular lattices. These lattices are explicitly constructed an d classified. Gram matrices of the lattices are given, using a discrete ana logue of Maslov index.