Symplectic group lattices
Rudolf Scharlau 
Department of Mathematics, University of Dortmund, 44221 Dortmund, Germany 
and Pham Huu Tiep 
Department of Mathematics, Ohio State University, Columbus, Ohio 43210 
Current address: Department of Mathematics, University of Florida, 
Gainseville, Florida 32611 
Trans. Amer. Math. Soc. 351 (1999), 2101-2139.

Abstract. 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 $\mathbb Q(\sqrt{\epsilon p})$,
where $\epsilon = (-1)^{(p-1)/2}$. We study the integral lattices related to 
these representations for the case $p^n \equiv \text{\rm mod} 4$. (The case 
$p^n \equiv 3 \text{\rm mod} 4$ has been considered in a previous
paper.) We show that the class of invariant lattices contains either 
unimodular or $p$-modular lattices. These lattices are explicitly constructed 
and classified. Gram matrices of the lattices are given, using a
discrete analogue of Maslov index.