Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (Annals of Mathematics Studies, 180) - Softcover

Katz, Nicholas M.

Softcover

ISBN 10: 0691153310 ISBN 13: 9780691153315

Publisher: Princeton University Press, 2012

Synopsis

Convolution and Equidistribution explores an important aspect of number theory--the theory of exponential sums over finite fields and their Mellin transforms--from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject.


The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book's main theorem, using a mixture of geometric, categorical, and group-theoretic methods.


By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject.

About this title

About the Author

Nicholas M. Katz is professor of mathematics at Princeton University. He is the author or coauthor of six previous titles in the Annals of Mathematics Studies: Arithmetic Moduli of Elliptic Curves (with Barry Mazur); Gauss Sums, Kloosterman Sums, and Monodromy Groups; Exponential Sums and Differential Equations; Rigid Local Systems; Twisted L-Functions and Monodromy; and Moments, Monodromy, and Perversity.

Excerpt. © Reprinted by permission. All rights reserved.

CONVOLUTION AND EQUIDISTRIBUTION

Sato-Tate Theorems for Finite-Field Mellin Transforms By Nicholas M. Katz

PRINCETON UNIVERSITY PRESS

Copyright © 2012 Princeton University Press
All right reserved.
ISBN: 978-0-691-15331-5

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

Overview

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 (G^m/k. In the question of Rudnick, we view (x + 1)/(x - 1) as a morphism from (G_m \ {1} to A¹, and form the lisse sheaf L_{ψ((x+1)/(x-1))} G_m \ {1}. With

j: G_m \ {1} -> G_m

the inclusion, we form the direct image sheaf j_* L_{ψ((x+1)/(x-1))} on (G_m/k (which for this sheaf, which is totally ramified at the point l, is the same as extending it by zero across the point 1).

The common feature of both questions is that we have a dense open set U/k [??] (G_m/k, a lisse, [??]-pure of weight zero sheaf F on U/k, its extension G := j_*F by direct image to (G_m/k, and we are looking at the sums

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To deal with the factor 1/[square root of q], we choose a square root of the l-adic unit p in [??]_l, and use powers of this chosen square root as our choices of [square root of q]. [For definiteness, we might choose that [square root of p] which via [??] becomes the positive square root, but either choice will do.] Because [square root of q] is an l-adic unit, we may form the "half"-rate twist G(1/2) of G, which for any finite extension field E/k and any point t [member of] (G_m(E) multiplies the traces of the Frobenii by 1/[square root of #E], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a final and apparently technical step, we replace the middle extension sheaf G(1/2) by the same sheaf, but now placed in degree -1, namely the object

M := G(1/2)[l]

in the derived category [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It will be essential in a moment that the object M is in fact a perverse sheaf, but for now we need observe only that this shift by one of the degree has the effect of changing the sign of each Trace term. In terms of this object, we are looking at the sums

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So written, the sums S(M, k x X) make sense for any object M [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].If we think of M as fixed but χ as variable, we are looking at the Metlin (:= multiplicative Fourier) transform of the function t [??] Trace (Frobt,k|M) on the finite abelian group G_m(k) = k^x. It is a standard fact that the Mellin transform turns multiplicative convolution of functions on k^x into multiplication of functions of χ.

On the derived category [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have a natural operation of !-convolution

(M, N) -> M *_! N

defined in terms of the multiplication map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the external tensor product object

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It then results from the Lefschetz Trace formula [Gr-Rat] and proper base change that, for any multiplicative character χ of k^x, we have the product formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

more generally, for any finite extension field Elk, and any multiplicative character ρ of E^x, we have the product formula

S(M *_! N, E, ρ) = S(M, E, ρ) S(N, E, ρ).

At this point, we must mention two technical points, which will be explained in detail in the next chapter, but which we will admit here as black boxes. The first is that we must work with perverse sheaves N satisfying a certain supplementary condition, P. This is the condition that, working on G_m/[??], N admits no subobject and no quotient object which is a (shifted) Kummer sheaf Lχ. For an N which is geometrically irreducible, P is simply the condition that N is not geometrically a (shifted) Kummer sheaf Lχ. Thus any geometrically irreducible N which has generic rank ≥ 2, or which is not lisse on (G_m, or which is not tamely ramified at both 0 and ∞, certainly satisfies P. Thus for example the object giving rise to the Evans sums, namely L_ψ(x-1/x)(1/2)[1], is wildly ramified at both 0 and ∞, and the object giving rise to the Rudnick sums, namely j_* L_{ψ((+1)/(x-1))} (1/2), is not lisse at 1 [member of] (G_m([??]), so both these objects satisfy P. The second technical point is that we must work with a variant of ! convolution *_!, called "middle" convolution *_mid, which is defined on perverse sheaves satisfying P, cf. the next chapter.

In order to explain the simple underlying ideas, we will admit four statements, and explain how to deduce from them equidistribution theorems about the sums S(M, k, χ) as χ varies.

(1) If M and N are both perverse on (G_m/k (resp. on (G_m/[??]) and satisfy P, then their middle convolution M *_mid N is perverse on (G_m/k (resp. on (G_m/[??]) and satisfies P.

(2) With the operation of middle convolution as the "tensor product," the skyscraper sheaf δ₁ as the "identity object," and [x [??] 1/x]^* DM as the "dual" M^V of M (DM denoting the Verdier dual of M), the category of perverse sheaves on (G^m/k (resp. on (G_m/[??]) satisfying P is a neutral Tannakian category, in which the "dimension" of an object M is its Euler characteristic χ_c((G_m/[??], M).

(3) Denoting by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the inclusion, the construction

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a fibre functor on the Tannakian category of perverse sheaves on (Gm/k satisfying P (and hence also a fibre functor on the subcategory of perverse sheaves on (G_m/k satisfying P). For i ≠ 0, Hⁱ (A1/[??]], j0!, M) vanishes.

(4) For any finite extension field Elk, and any multiplicative character ρ of E^x, the construction

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is also such a fibre functor. For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] vanishes.

Now we make use of these four statements. Take for N a perverse sheaf on (G_m/k which is [??]-pure of weight zero and which satisfies P. Denote by <N>_arith the full subcategory of all perverse sheaves on (G_m/k consisting of all subquotients of all "tensor products" of copies of N and its dual N^V. Similarly, denote by <N>_geom the full subcategory of all perverse sheaves on (G_m/[??] consisting of all subquotients, in this larger category, of all "tensor products" of copies of N and its dual N^V. With respect to a choice ω of fibre functor, the category <N>_arith becomes the category of finite-dimensional [??]_l-representations of an algebraic group Garith,N,ω [??] GL(ω (N)) = GL ("dim " N), with N it-self corresponding to the given "dim" N-dimensional representation. Concretely, Garith,N,ω [??] GL(ω(N)) is the subgroup consisting of those automorphisms γ of ω(N) with the property that γ, acting on ω(M), for M any tensor construction on ω(N) and its dual, maps to itself every vector space subquotient of the form ω(any subquotient of M).

And the category <N>_geom becomes the category of finite-dimensional [[??]_l-representations of a possibly smaller algebraic group G_geom,N,ω [??] G_arith,N,ω (smaller because there are more subobjects to be respected).

For ρ a multiplicative character of a finite extension field E/k, we have the fibre functor ωρ defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

on <N>_arith. The Frobenius Frob_E is an automorphism of this fibre functor, so defines an element Frob_E,ρ in the group G_arith,N,ωρ defined by this choice of fibre functor. But one knows that the groups G_arith,N,ω (respectively the groups G_geom,N,ω) defined by different fibre functors are pairwise isomorphic, by a system of isomorphisms which are unique up to inner automorphism of source (or target). Fix one choice, say ω₀, of fibre functor, and define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the element Frob_E,ρ in the group G_arith,N,ωρ still makes sense as a conjugacy class in the group G_arith,N.

Let us say that a multiplicative character ρ of some finite extension field E/k is good for N if, for

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the inclusion, the canonical "forget supports" map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an isomorphism. If ρ is good for N, then the natural "forget supports" maps

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

together with the restriction map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are all isomorphisms. Moreover, as N is [??]-pure of weight zero, each of these groups is [??]-pure of weight zero.

Conversely, if the group [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is [??]-pure of weight zero, then ρ is good for N, and we have a "forget supports" isomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This criterion, that ρ is good for N if and only if ω_ρ(N) is [??]-pure of weight zero, shows that if ρ is good for N, then ρ is good for every object M in the Tannakian category <N>_arith generated by N, and hence that for any such M, we have an isomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(Continues...)

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Edition details

• Publisher: Princeton University Press
• Publication date: 2012
• Language: English
• ISBN 10: 0691153310
• ISBN 13: 9780691153315
• Binding: Paperback
• Number of pages: 208