Finite monodromy of some families of exponential sums

We first develop some basic facts about hypergeometric sheaves on the multiplicative group G m in characteristic p > 0. Certain of their Kummer pullbacks extend to irreducible local systems on the affine line in characteristic p > 0. One of these, of rank 23 in characteristic p = 3, turns out to have the Conway group Co 2 , in its irreducible orthogonal representation of degree 23, as its arithmetic and geometric monodromy groups.

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

Katz¹,

Rojas‐León²,

Tiep³

2019

Trans. Amer. Math. Soc.

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In the first three sections, we develop some basic facts about hypergeometric sheaves on the multiplicative group 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.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.

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Finite monodromy of some families of exponential sums

Abstract: Given a prime p and an integer d > 1, we give a numerical criterion to decide whether the ℓ-adic sheaf associated to the one-parameter exponential sums t → x ψ(x d + tx) over Fp has finite monodromy or not, and work out some explicit cases where this is computable.

Cited by 4 publications

References 10 publications

A rigid local system with monodromy group 2.J2

A rigid local system with monodromy group 2.J2

Rigid local systems with monodromy group the Conway group Co3

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

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