A rigid local system with monodromy group the big Conway group 2.Co_1 and two others with monodromy group the Suzuki group 6.Suz

Abstract

In the first three sections, we develop some basic facts about hypergeometric sheaves on the multiplicative group ${\mathbb G}_m$ in characteristic $p >0$. In the fourth and fifth sections, we specialize to quite special classses of hypergeomtric sheaves. We give relatively "simple" formulas for their trace functions, and a criterion for them to have finite monodromy. In the next section, we prove that three of them have finite monodromy groups.We then give some results on finite complex linear groups. We next use these group theoretic results to show that one of our local systems, of rank $24$ in characteristic $p=2$, has the big Conway group $2.\mathrm{Co}_1$, 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 $6.\mathrm{Suz}$, 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. In the final section, we pull back these local systems by $x \mapsto x^N$ maps to ${\mathbb A}^1$, and show that after pullback their arithmetic and geometric monodromy groups remain the same. Sadly the Leech lattice makes no appearance in our arguments.