2019/01/12 by Nicholas M. Katz, Katz, Nicholas M., Antonio Rojas‐León +3
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1901.03894
openalex publication_date 2019/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the first three sections, we develop some basic facts about hypergeometric\nsheaves on the multiplicative group mathbb Gm in characteristic p >0.\nIn the fourth and fifth sections, we specialize to quite special classses of\nhypergeomtric sheaves. We give relatively "simple" formulas for their trace\nfunctions, and a criterion for them to have finite monodromy. In the next\nsection, we prove that three of them have finite monodromy groups.We then give\nsome results on finite complex linear groups.\n We next use these group theoretic results to show that one of our local\nsystems, of rank 24 in characteristic p=2, has the big Conway group\n2.\Co1, in its irreducible orthogonal representation of degree 24\nas the automorphism group of the Leech lattice, as its arithmetic and geometric\nmonodromy groups. Each of the other two, of rank 12 in characteristic p=3,\nhas the Suzuki group 6.\Suz, in one of its irreducible\nrepresentations of degree 12 as the mathbb Q(\ζ3)-automorphisms of\nthe Leech lattice, as its arithmetic and geometric monodromy groups. In the\nfinal section, we pull back these local systems by x \↦ xN maps to\n mathbb A1, and show that after pullback their arithmetic and geometric\nmonodromy groups remain the same. Sadly the Leech lattice makes no appearance\nin our arguments.\n