A rigid local system with monodromy group the big Conway group $2.\mathsf {Co}_1$ and two others with monodromy group the Suzuki group $6.{\sf {Suz}}$

Abstract:

We first develop some basic facts about hypergeometric sheaves on the multiplicative group $\mathbb {G}_m$ in characteristic $p>0$. Specializing to some special classes of hypergeometric sheaves, we give relatively “simple" formulas for their trace functions, and a criterion for them to have finite monodromy. We then show that one of our local systems, of rank $24$ in characteristic $p=2$, has the big Conway group $2.\mathsf {Co}_1$, in its irreducible orthogonal representation of degree $24$ as the automorphism group of the Leech lattice, as its arithmetic and geometric monodromy groups. Each of the other two, of rank $12$ in characteristic $p=3$, has the Suzuki group $6.\mathsf {Suz}$, in one of its irreducible representations of degree $12$ as the $\mathbb {Q}(\zeta _3)$-automorphisms of the Leech lattice, as its arithmetic and geometric monodromy groups. We also show that the pullback of these local systems by $x \mapsto x^N$ mappings to the affine line $\mathbb {A}^1$ yields the same arithmetic and geometric monodromy groups.