A rigid local system with monodromy group 2.J2

Abstract

We exhibit a rigid local system of rank six on the affine line in characteristic p = 5 whose arithmetic and geometric monodromy groups are the finite group 2.J 2 (J 2 the Hall-Janko sporadic group) in one of its two (Galois-conjugate) irreducible representation of degree six.

Finite monodromy of some families of exponential sums

“…Studying the monodromy group of these sheaves is interesting for its applications to coding theory and also as they give interesting examples of lisse sheaves whose monodromy group is a sporadic simple finite group [KR19]. In [Roj19] it was shown that, if f (x) is a monomial, then the monodromy is either finite, the full symplectic group Sp d−1 (for d odd) or the special linear group SL d−1 (for d even) (more generally, see [KT24, Theorem 10.2.4]).…”