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Convolution and Equidistribution by Nicholas M. Katz

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CHAPTER 22

SL(n)Examples with Slightly Composite n

In this chapter, we continue to study the object N of Theorem 21.1. Thus k is a finite field of characteristic p, Ã a nontrivial additive character of k, f(x) = ∑ai=–b Aixi ∈ k[x, 1/x] is a Laurent polynomial of “bidegree” (a, b), with a, b both 1 and both prime to p. We assume that f(x) is Artin-Schreier reduced. We take for N the object N := LÃ(f(x))(1/2)[1] ∈ Parith.

Theorem 22.1. The object N, viewed in <N>geom, is not geometrically isomorphic to the middle convolution of any two objects K and L in <N>geom each of which has dimension ≥ 2. Equivalently, the representation of Ggeom,N corresponding to N is not the tensor product of two other representations of Ggeom,N, each of which has dimension ...

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