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Convolution and Equidistribution by Nicholas M. Katz

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CHAPTER 14

The Case of SL(2); the Examples of Evans and Rudnick

In treating both of these examples, as well as all the examples to come, we will use the Euler-Poincaré formula, cf. [Ray, Thm. 1] or [Ka-GKM, 2.3.1] or [Ka-SE, 4.6, (v) atop p. 113] or [De-ST, 3.2.1], to compute the “dimension” of the object N in question.

Let us briefly recall the general statement of the Euler-Poincaré formula, and then specialize to the case at hand. Let X be a projective, smooth, nonsingular curve over an algebraically closed field k in which is invertible, U ⊂ X a dense open set in X, and V ⊂ U a dense open set in U. Let G be a constructible Q-sheaf on U which is lisse on V of rank r := gen.rk.(G). We view G|V as a representation of π1(V). For each point ...

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