Symplectic group lattices

Scharlau, Rudolf; Tiep, Pham

doi:10.1090/S0002-9947-99-02469-1

Symplectic group lattices
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by Rudolf Scharlau and Pham Huu Tiep PDF

Trans. Amer. Math. Soc. 351 (1999), 2101-2139 Request permission

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 $p^{n} \equiv 1 \bmod 4$. (The case $p^{n} \equiv 3 \bmod 4$ 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.

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