Über die Autorin bzw. den Autor

Nicholas M. Katz

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CONVOLUTION AND EQUIDISTRIBUTION

PRINCETON UNIVERSITY PRESS

Contents

Introduction......................................................................................1Chapter 1. Overview...............................................................................7Chapter 2. Convolution of Perverse Sheaves........................................................19Chapter 3. Fibre Functors.........................................................................21Chapter 4. The Situation over a Finite Field......................................................25Chapter 5. Frobenius Conjugacy Classes............................................................31Chapter 6. Group-Theoretic Facts about Ggeom and Garith.....................33Chapter 7. The Main Theorem.......................................................................39Chapter 8. Isogenies, Connectedness, and Lie-Irreducibility.......................................45Chapter 9. Autodualities and Signs................................................................49Chapter 10. A First Construction of Autodual Objects..............................................53Chapter 11. A Second Construction of Autodual Objects.............................................55Chapter 12. The Previous Construction in the Nonsplit Case........................................61Chapter 13. Results of Goursat-Kolchin-Ribet Type.................................................63Chapter 14. The Case of SL(2); the Examples of Evans and Rudnick..................................67Chapter 15. Further SL(2) Examples, Based on the Legendre Family..................................73Chapter 16. Frobenius Tori and Weights; Getting Elements of Garith.....................77Chapter 17. GL(n) Examples........................................................................81Chapter 18. Symplectic Examples...................................................................89Chapter 20. GL(n) x GL(n) x ... x GL(n) Examples..................................................113Chapter 21. SL(n) Examples, for n an Odd Prime....................................................125Chapter 22. SL (n) Examples with Slightly Composite n.............................................135Chapter 23. Other SL (n) Examples.................................................................141Chapter 24. An O(2n) Example......................................................................145Chapter 25. G2 Examples: the Overall Strategy..........................................147Chapter 26. G2 Examples: Construction in Characteristic Two............................155Chapter 27. G2 Examples: Construction in Odd Characteristic............................163Chapter 28. The Situation over Z: Results.........................................................173Chapter 29. The Situation over Z: Questions.......................................................181Chapter 30. Appendix: Deligne's Fibre Functor.....................................................187Bibliography......................................................................................193

Chapter One

Let k be a finite field, q its cardinality, p its characteristic,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a nontrivial additive character of k, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a (possibly trivial) multiplicative character of k.

The present work grew out of two questions, raised by Ron Evans and Zeev Rudnick respectively, in May and June of 2003. Evans had done numerical experiments on the sums

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as χ varies over all multiplicative characters of k. For each χ, S(χ) is real, and (by Weil) has absolute value at most 2. Evans found empirically that, for large q = #k, these q - 1 sums were approximately equidistributed for the "Sato-Tate measure" (1/2π) [square root of 4 - x²]dx on the closed interval [-2, 2], and asked if this equidistribution could be proven.

Rudnick had done numerical experiments on the sums

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as χ varies now over all nontrivial multiplicative characters of a finite field k of odd characteristic, cf. [KRR, Appendix A] for how these sums arose. For nontrivial χ, T(χ) is real, and (again by Weil) has absolute value at most 2. Rudnick found empirically that, for large q = #k, these q-2 sums were approximately equidistributed for the same "Sato-Tate measure" (1/2π) [square root of 4 - x²]dx on the closed interval [-2,2], and asked if this equidistribution could be proven.

We will prove both of these equidistribution results. Let us begin by slightly recasting the original questions. Fixing the characteristic p of k, we choose a prime number l [≠] p; we will soon make use of l-adic etale cohomology. We denote by Zl the l-adic completion of Z, by Q_l its fraction field, and by [??]_l an algebraic closure of Q_l. We also choose a field embedding [??] of [??]_l into (C. Any such [??] induces an isomorphism between the algebraic closures of Q in [??]_l and in (C respectively. By means of [??], we can, on the one hand, view the sums S(χ) and T(χ) as lying in [??]_l. On the other hand, given an element of [??]_l, we can ask if it is real, and we can speak of its complex absolute value. This allows us to define what it means for a lisse sheaf to be [??]-pure of some weight w (and later, for a perverse sheaf to be [??]-pure of some weight w). We say that a perverse sheaf is pure of weight w if it is [??]-pure of weight w for every choice of [??].

By means of the chosen [??], we view both the nontrivial additive character ψ of k and every (possibly trivial) multiplicative character χ X of k^x as having values in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, attached to ψ we have the Artin-Schreier sheaf L_ψ = L_ψ(x) on A¹/k := Spec(k[x]), a lisse [??]_l-sheaf of rank one on A₁/k which is pure of weight zero. And for each X we have the Kummer sheaf [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], a lisse [??]_l-sheaf of rank one on (G_m k which is pure of weight zero. For a k-scheme X and a k-morphism f : X -> A1/k (resp. f : X -> G_m/k), we denote by L_ψ(f) (resp. L_χ(f)) the pullback lisse rank one, pure of weight zero, sheaf f^* L_ψ(f) (resp. f^* L_χ(f)) on X.

In the question of Evans, we view x - 1/x as a morphism from (Gm to A¹, and form the lisse sheaf L_ψ(x-1/x) on...