Convolution and Equidistribution

Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Nicholas M. Katz
Copyright Date: January 2012
Pages: 212
https://www.jstor.org/stable/j.ctt7svh4
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    Convolution and Equidistribution
    Book Description:

    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.

    eISBN: 978-1-4008-4270-4
    Subjects: Mathematics, Statistics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-viii)
  3. Introduction
    (pp. 1-6)

    The systematic study of character sums over finite fields may be said to have begun over 200 years ago, with Gauss. The Gauss sums over $ \mathbb{F} _p $ are the sums

    $ \displaystyle\sum _{x \in \mathbb{F}_p^\times} \, \psi (x) \chi (x), $

    for ψ a nontrivial additive character of $ \mathbb{F} _p $ , e.g., xe²πix/p, and χ a nontrivial multiplicative character of $ \mathbb{F} _p^\times $ . Each has absolute value $ \sqrt{p} $ . In 1926, Kloosterman [Kloos] introduced the sums (one for each $ a \in \mathbb{F} _p^\times $ )

    $ \displaystyle\sum _{xy=a \quad \text{in} \quad \mathbb{F} \text{p}} \quad \psi (x \, + \, y) $

    which bear his name, in applying the circle method to the problem of four squares. In 1931 Davenport [Dav] became interested in (variants of) the following questions: for how...

  4. CHAPTER 1 Overview
    (pp. 7-18)

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

    $ \psi \, : \, (k, \, +) \, \rightarrow \, \mathbb{Z} [\zeta_p]^\times \, \subset \, \mathbb{C} ^\times $

    a nontrivial additive character of k, and

    $ \chi \, : \, (k ^\times, \, \times) \, \rightarrow \, \mathbb{Z} [\zeta _{q-1}]^\times \, \subset \, \mathbb{C}^\times $

    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

    $ S(\chi) \, := \, -(1/\sqrt{q}) \displaystyle \sum _{t \in k ^\times} \quad \psi (t - 1/t) \chi (t) $

    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...

  5. CHAPTER 2 Convolution of Perverse Sheaves
    (pp. 19-20)

    Let k be a finite field, q its cardinality, p its characteristic, $ \ell \, \neq \, p $ a prime number, and G/k a smooth commutative groupscheme which over $ \overline k $ becomes isomorphic to $ \mathbb{G} _m/\overline k $ . We will be concerned with perverse sheaves on G/k and on $ G/\overline k $ .

    We begin with perverse sheaves on $ G/\overline k \, \cong \, \mathbb{G} _m/\overline k $ . On the derived category $ D _c^b (\mathbb{G}_m/k, \, \mathbb{\overline Q}_\ell) $ we have two notions of convolution, ! convolution and * convolution, defined respectively by

    $ N \, \star_! \, M \, := \, R \pi_!(N \, \boxtimes \, M), $

    $ N \, \star_\star \, M \, := \, R \pi_\star (N \, \boxtimes \, M), $

    where $ \pi = \mathbb{G}_m \, \times \, \mathbb{G}_m \, \rightarrow \, \mathbb{G}_m $ is the multiplication map. For neither of these notions is it the case that the convolution of two perverse sheaves need be perverse. In our...

  6. CHAPTER 3 Fibre Functors
    (pp. 21-24)

    At this point, we introduce the fibre functor suggested by Deligne. The proof that it is in fact a fibre functor is given in the Appendix.

    Theorem 3.1. (Deligne) Denoting by

    $ j_0 \, : \, \mathbb{G}_m/\overline k \, \subset \, \mathbb{A} ^1/\overline k $

    the inclusion, the construction

    $ M \, \mapsto \, H^0 ( \mathbb{A}^1 / \overline k, \, j_{0!} M) $

    is a fibre functor on the Tannakian category $ \mathcal{P} $ .

    For any Kummer sheaf $ \mathcal{L}_\chi $ on $ \mathbb{G}_m/\overline k $ , the operation $ M \, \mapsto \, M \, \otimes \, \mathcal{L}_\chi $ is an autoequivalence of $ \mathcal{P} $ with itself as Tannakian category. So we get the following corollary.

    Corollary 3.2. For any Kummer sheaf $ \mathcal{L}_\chi $ on $ \mathbb{G}_m/\overline k $ , the construction

    $ M \, \mapsto \, H^0(\mathbb{A}^1 / \overline{k}, j_{0!}(M \, \otimes \, \cal{L}_\chi)) $

    is a fibre functor ωχ on the Tannakian category $ \cal{P} $ .

    Let us say...

  7. CHAPTER 4 The Situation over a Finite Field
    (pp. 25-30)

    Let us now turn our attention to the case of a finite field k, and a groupscheme G/k which is a form of $ \mathbb{G}_m $ . Concretely, it is either $ \mathbb{G}_m/\overline k $ itself, or it is the nonsplit form, defined in terms of the unique quadratic extension k₂/k inside the chosen $ \overline k $ as follows: for any k-algebra A,

    $ G(A) \, := \, \{x \, \in \, A \, \otimes_k \, k_2 | {\text \, {Norm}} \,_{A \otimes_k k_2} (x) \, = \, 1 \}. $

    We denote by $ \mathcal{P}_arith $ the full subcategory of the category Pervarith of all perverse sheaves on G/k consisting of those perverse sheaves on G/k which, pulled back to $ G/\overline k $ , lie in $ \mathcal {P} $ . And we denote by Negarith the full subcategory of...

  8. CHAPTER 5 Frobenius Conjugacy Classes
    (pp. 31-32)

    Let G/k be a form of $ \mathbb{G}_m $ , and N in $ \mathcal{P}_arith $ an object which is ι-pure of weight zero and arithmetically semisimple. If G/k is $ \mathbb{G}_m $ , then we have the fibre functor on <N>arith given by

    $ M \, \mapsto \, \omega (N) \, := \, H^0 (\mathbb{A} ^1/\overline k, \, j_{0!}M), $

    on which Frobk operates. And for each finite extension field E/k and each character χ of G(E), we have the fibre functor ωχ on <N>arith given by

    $ \omega_\chi (M) \, := \, \omega (M \, \otimes \, \mathcal{L}_\chi), $

    on which FrobE operates. This action of FrobE on ωχ gives us an element in the Tannakian group Garith,N,ωχ for <N>arith, and so a conjugacy class FrobE,χ in the reference Tannakian group Garith,N...

  9. CHAPTER 6 Group-Theoretic Facts about Ggeom and Garith
    (pp. 33-38)

    Theorem 6.1. Suppose N in $ \mathcal{P}_arith $ is geometrically semisimple. Then Ggeom,N is a normal subgroup of Garith,N.

    Proof. Because N is geometrically semisimple, the group Ggeom,N is reductive, so it is the fixer of its invariants in all finite dimensional representations of the ambient Garith,N. By noetherianity, there is a finite list of representations of Garith,N such that Ggeom,N is the fixer of its invariants in these representations. Taking the direct sum of these representations, we get a single representation of Garith,N such that Ggeom,N is the fixer of its invariants in that single representation. This representation corresponds to an object...

  10. CHAPTER 7 The Main Theorem
    (pp. 39-44)

    Lemma 7.1. Let G/k be a form of $ \mathbb{G}_m $ , and N in $ \mathcal{P}_arith $ ι-pure of weight zero and arithmetically semisimple. The quotient group Garith,N/Ggeom,N is a group of multiplicative type, in which a Zariski dense subgroup is generated by the image of any single Frobenius conjugacy class Frobk,χ. If the quotient is finite, say of order n, then it is canonically $ \mathbb{Z}/n\mathbb{Z} $ , and the image in this quotient of any Frobenius conjugacy class FrobE,χ is deg(E/k) mod n.

    Proof. Representations of the quotient Garith,N/Ggeom,N are objects in <N>arith which are geometrically trivial, i.e., those objects $ V \otimes \, \delta _1 $ for V...

  11. CHAPTER 8 Isogenies, Connectedness, and Lie-Irreducibility
    (pp. 45-48)

    For each prime to p integer n, we have the n’th power homomorphism [n] : GG. Formation of the direct image

    $ M \, \mapsto \, [n]_\star M $

    is an exact functor from Perv to itself, which maps Neg to itself, $ \mathcal{P} $ to itself, and which (because a homomorphism) is compatible with middle convolution:

    $ [n]_\star(M \, \star_{mid} \, N) \, \cong \, ([n]_\star M) \,\star_{mid} ([n]_\star N). $

    So for a given object N in $ \mathcal{P}_arith $ , [n]* allows us to view <N>arith as a Tannakian subcategory of <[n]*N>arith, and <N>geom as a Tannakian subcategory of <[n]*N>geom. For the fibre functor ω defined (after a choice of isomorphism $ G/\overline{k} \, \cong \, \mathbb{G}_m/\overline{k} $ ) by

    $ N \mapsto \omega(N) \, := \, H^0 (\mathbb{A}^1/\overline{k}, \, j_{0!} N), $

    we have canonical functorial isomorphisms...

  12. CHAPTER 9 Autodualities and Signs
    (pp. 49-52)

    Suppose that N in $ \mathcal{P}_arith $ is geometrically irreducible (so a fortiori arithmetically irreducible) and ι-pure of weight zero. Suppose further that N is arithmetically self-dual in $ \mathcal{P}_arith $ , i.e., that there is an arithmetic isomorphism N ≅ [x ↦ 1/x]*DN, DN denoting the Verdier dual of N. This arithmetic isomorphism is then unique up to a scalar factor. It induces an autoduality on ω(N) which is respected by Garith,N. Up to a scalar factor, this is the unique autoduality on ω(N) which is respected by Garith,N, so it is either an orthogonal or a symplectic autoduality. We say that the...

  13. CHAPTER 10 A First Construction of Autodual Objects
    (pp. 53-54)

    These constructions are based on evaluating the sum

    $ (1/ \# G (E)) \quad \displaystyle \sum _{\rho \in \text{Good} (E, N)} \text{Trace} (Frob^2_{E,\rho} | \omega (N)) $

    $ =(1/ \# G (E)) \quad \displaystyle \sum _{\rho \in \text{Good} (E, N)} \text{Trace} (Frob_E_2 | H^0_c (G / \overline k, N \, \otimes \, \cal L_\rho)) $

    more or less precisely. As always, this sum is within O(1/#E) of the sum

    $ (1/ \# G (E)) \quad \displaystyle \sum _{all \; \rho \in G (E)^\vee} \text{Trace} (Frob_E_2 | H^0_c (G / \overline k, N \, \otimes \, \cal L_\rho)), $

    which is in turn equal, by the Lefschetz Trace formula [Gr-Rat], to

    $ (1/ \# G (E)) \quad \displaystyle \sum _{all \; \rho \in G (E)^\vee} \, \displaystyle \sum _{t \in G (E_2)} \rho (\text{Norm} _{E_2/E} (t)) \text{Trace} (Frob_{E_{2,t}} | N). $

    This sum, by orthogonality, is

    $ \displaystyle \sum _{t \in G (E_2) | \text{Norm} _{E_2/E}(t) = 1} \text{Text} (Frob _{E _{2, t}} | N). $

    We begin with a geometrically irreducible middle extension sheaf $ \cal{F} $ on G/k which is τ-pure of weight zero, and which is not geometrically an $ \cal{L}_\chi $ . Thus $ \cal{F}(1/2)[1] $ is a geometrically irreducible object in $ \cal{P}_{arith} $ . Its dual in $ \cal{P}_{arith} $ is $ [x \, \mapsto \, 1/x]* \cal \overline {F} (1/2) [1] $ , for $ \cal \overline{F} $ the linear dual middle extension sheaf. Via τ, $ \cal{F} $ and...

  14. CHAPTER 11 A Second Construction of Autodual Objects
    (pp. 55-60)

    In this construction, we work on the split form $ \mathbb{G} _m/k, \, Spec (k[x, \, 1/x]) $ . We begin with a geometrically irreducible lisse sheaf $ \cal F $ on an open dense set $ U \subset \mathbb{G} _m $ which is ι-pure of weight zero and which is self-dual: $ \cal F \, \cong \, \cal \overline F $ .

    Denote by d the rank of $ \cal F $ . We view $ \cal{F} | U $ as a d-dimensional representation ρ of $ \pi ^{arith}_1 (U) $ , toward either the orthogonal group $ O(d)/\mathbb \overline {Q_\ell} $ , if the autoduality is orthogonal, or toward the symplectic group $ Sp(d)/\mathbb \overline{Q_\ell} $ if the autoduality is symplectic (which forces d to be even). We denote by $ G_{geom,\cal{F}} $ the Zariski closure of the image $ \rho(\pi^{geom}_1(U)) $ of the geometric...

  15. CHAPTER 12 The Previous Construction in the Nonsplit Case
    (pp. 61-62)

    In this construction, we work on the nonsplit form G/K. Denoting by k₂/k the unique quadratic extension inside $ \overline k $ , recall that for any k-algebra A, G(A) is the group of elements $ t \, \in \, (A \, \otimes _k \, k_2)^\times $ with $ \text{Norm} _{A\otimes_k k_2/A} (t) = 1 $ . Thus $ G(A) \subset A \, \otimes _k \, k_2 $ . We have the trace map $ \text{Trace} _{A\otimes_kk_2/A} \, : \, A \, \otimes_k \, k_2 \, \rightarrow \, A $ . Restricting it to G(A), we get, for any k-algebra A, a map Trace : G(A) → A, in other words a k-morphism $ \text{Trace} \, : \, G/k \rightarrow \mathbb{A} ^1/k $ , i.e., a function Trace on G/k.

    A basic observation is that the function Trace is invariant under inversion: for $ t \, \in \, G(A) $ , Trace(t) = Trace(1/t). Indeed for $ t \, \in \, G(A) $ , we claim...

  16. CHAPTER 13 Results of Goursat-Kolchin-Ribet Type
    (pp. 63-66)

    Suppose we are given some number r ≥ 2 of objects N₁, N₂, …, Nr in $ \cal P_{arith} $ of some common “dimension” d ≥ 1. Suppose they are all ι-pure of weight zero, geometrically irreducible, and arithmetically self-dual, all with the same sign of duality.

    Theorem 13.1. Suppose that d ≥ 2 is even, that each Ni is symplectically self-dual, and that for each i = 1, …, r, we have Ggeom,Ni = Garith,Ni = Sp(d). Suppose further that for ij, there is no geometric isomorphism between Ni and Nj and there is no geometric isomorphism between Ni and [x...

  17. CHAPTER 14 The Case of SL(2); the Examples of Evans and Rudnick
    (pp. 67-72)

    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 briey 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 $ \overline k $ in which is invertible, UX a dense open set in X, and VU a dense...

  18. CHAPTER 15 Further SL(2) Examples, Based on the Legendre Family
    (pp. 73-76)

    In this chapter, we suppose that k has odd characteristic. We begin with the Legendre family of elliptic curves over the λ line, given in $ \mathbb{P}^2 \, \times \, \mathbb{A}^1 $ by the equation

    $ Y^2Z = X(X-Z)(X-\lambda Z). $

    For π its projection onto $ \mathbb{A}^1 $ , we define

    $ Leg \, := \, R^1 \pi_! \mathbb \overline{Q_\ell}. $

    Thus Leg is lisse of rank two and pure of weight one (Hasse’s theorem [Ha-Ell, page 205]) outside of 0 and 1. For $ j \, : \, \mathbb{A} ^1 \, \setminus \, \{0, \, 1\} \, \subset \, \mathbb{A} ^1 $ the inclusion we have Leg = j*j*Leg. One knows [De-Weil II, 3.5.5] that the geometric monodromy group of the lisse sheaf j*Leg is SL(2). Its local monodromy at both 0 and 1 is a single unipotent...

  19. CHAPTER 16 Frobenius Tori and Weights; Getting Elements of Garith
    (pp. 77-80)

    In this chapter, we work on $ \mathbb{G}_m/k $ . We consider an arithmetically semisimple object $ N \in \cal {P}_{arith} $ which is pure of weight zero. We assume it is of the form $ \cal {G}[1] $ , with $ \cal{G} $ a middle extension sheaf. Thus for some open set $ j \, : \, U \, \subset \, \mathbb{G}_m $ , we have $ \cal{G} = j_*\cal{F} $ for $ \cal{F} \, := j^*\cal{G} $ a lisse sheaf on U which is pure of weight −1 and arithmetically semisimple, and having no geometric constituent isomorphic to (the restriction to $ U _{\overline k} $ a Kummer sheaf. Recall that Deligne’s fibre functor is (for $ j_0 \, : \, \mathbb{G}_m \, \subset \, \mathbb{A}^1 $ the inclusion)

    $ M \, \mapsto \, \omega(N) \, := \, H^0(\mathbb{A}^1 \, \otimes_k \overline k, j_{0!} N). $

    The action of Frobk on the restriction to <N>arith of this fibre functor...

  20. CHAPTER 17 GL(n) Examples
    (pp. 81-88)

    Here we work on either the split or the nonsplit form. We begin with a lisse sheaf $ \cal{F} $ on a dense open set j : UG which is geometrically irreducible, pure of weight zero, and not geometrically isomorphic to (the restriction to U of) any Kummer sheaf $ \cal{L}_\chi $ . We denote by $ \cal{G} \, := j_*\cal{F} $ its middle extension to G. Then the object $ N \, := \cal{G} (1/2)[1] \, \in \, \cal{P}_{arith} $ is pure of weight zero and geometrically irreducible.

    Theorem 17.1. Suppose that N is not geometrically isomorphic to any nontrivial multiplicative translate of itself. Suppose further that for one of the two possible geometric isomorphisms $ G/\overline{k} \, \cong \, \mathbb{G}_m/\overline{k}, \, \cal{F}(0)^{unip} $ ...

  21. CHAPTER 18 Symplectic Examples
    (pp. 89-102)

    We work on either the split or the nonsplit form. We begin with a lisse sheaf $ \cal{F} $ on a dense open set j : UG which is geometrically irreducible, pure of weight zero, and not geometrically isomorphic to (the restriction to U of) any Kummer sheaf $ \cal{L}_\chi $ . We denote by $ \cal{G} \, := \, j_*\cal{F} $ its middle extension to G. Then the object $ N \, := \cal{G}(1/2)[1] \, \in \, \cal{P}_{arith} $ is pure of weight zero and geometrically irreducible.

    Theorem 18.1. Suppose that N is not geometrically isomorphic to any nontrivial multiplicative translate of itself, and that N is symplectically self-dual. Suppose further that for either of the...

  22. CHAPTER 19 Orthogonal Examples, Especially SO(n) Examples
    (pp. 103-112)

    The orthogonal case is more difficult than the symplectic one because of the need to distinguish between SO(n) and O(n), which we do not in general know how to do. We work on either the split or the nonsplit form. We begin with a lisse sheaf $ \cal{F} $ on a dense open set j : UG which is geometrically irreducible, pure of weight zero, and not geometrically isomorphic to (the restriction to U of) any Kummer sheaf $ \cal{L}_\chi $ . We denote by $ \cal{G} \, := j_*\cal{F} $ its middle extension to G. Then the object $ N \, := \, \cal{G}(1/2)[1] \, \in \, \cal{P}_{arith} $ is pure of weight zero and geometrically...

  23. CHAPTER 20 GL(n) × GL(n) × … × GL(n) Examples
    (pp. 113-124)

    In this chapter, we investigate the following question. Suppose we have a geometrically irreducible middle extension sheaf $ \cal{G} $ on $ \mathbb {G}_m/k $ which is pure of weight zero, such that the object $ N \, := \, \cal{G} (1/2)[1] \, \in \, \cal{P} _{arith} $ has “dimension” n and has Ggeom,N = Garith,N = GL(n). Suppose in addition we are given s ≥ 2 distinct characters χi of k×. We want criteria which insure that for the objects

    $ N_i \,:= \, N \, \otimes \, \cal{L} _{\chi}_i, $

    the direct sum ⊕iNi has $ G_{geom,\oplus_iN_i} \, = \, G_{arith,\oplus_iN_i} \, = \, \prod_i GL(n) $ . Because we have a priori inclusions $ G_{geom,\oplus_iN_i} \, \subset \, G_{arith,\oplus_iN_i} \, \subset \, \prod_i GL(n) $ , it suffices to prove that $ G_{geom,\oplus_iN_i} \, = \, \prod_i GL(n) $ . To show this, it suffices to show both of the following two statements....

  24. CHAPTER 21 SL(n) Examples, for n an Odd Prime
    (pp. 125-134)

    In this chapter, we will construct, for every n ≥ 3, n-dimensional objects with GgeomGarithSL(n), but only when n is prime will we be able to prove that Ggeom = Garith = SL(n).

    Theorem 21.1. Let k be a finite field of characteristic p, ψ a nontrivial additive character of k. Let $ f(x) = \sum^a_{i=-b} \, A_ix^i \, \in \, k[x, \, 1/x] $ be a Laurent polynomial of “bidegree” (a, b), with a, b both ≥ 1 and both prime to p. Assume further that f(x) is Artin-Schreier reduced. Thus AaA−b ≠ 0, and Ai ≠ 0 implies that i is prime to p. We have the...

  25. CHAPTER 22 SL(n) Examples with Slightly Composite n
    (pp. 135-140)

    In this chapter, we continue to study the object N of Theorem 21.1. Thus k is a finite field of characteristic p, ψ a nontrivial additive character of k, $ f(x) \, = \, \sum^a_{i=-b} \, A_ix^i \, \in \, 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 \, := \, \cal{L}_{\psi(f(x))} (1/2)[1] \, \in \, \cal{P}_{arith} $ .

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

  26. CHAPTER 23 Other SL(n) Examples
    (pp. 141-144)

    In this chapter, we fix an integer n ≥ 3 which is not a power of the characteristic p, and a monic polynomial $ f(x) \, \in \, k[x] $ of degree n, $ f(x) \, = \, \sum^n_{i=0} \,A_ix^i $ , An = 1.

    Lemma 23.1. Suppose that f has n distinct roots in $ \overline k $ , all of which are nonzero (i.e., A₀ ≠ 0). Let χ be a nontrivial character of k× with $ \chi^n \, = \, \mathbb{1} $ . Form the object $ N \,:= \, \cal{L}_{\chi(f)} (1/2)[1] \, \in \, \cal{P}_{arith} $ . Its Tannakian determinant “det” (N) is geometrically of finite order. It is geometrically isomorphic to δa for a = (-1)n A₀ = (-1)n f(0) = the product of all the zeroes of f.

    Proof. Let ρ...

  27. CHAPTER 24 An O(2n) Example
    (pp. 145-146)

    In this chapter, we work over a finite field k of odd characteristic. Fix an even integer 2n ≥ 4 and a monic polynomial $ f(x) \, \in \, k[x] $ of degree 2n, $ f(x) \, = \, \sum^{2n}_{i=0} \, A_ix^i $ , An = 1. We make the following three assumptions about f.

    (1) f has 2n distinct roots in $ \overline k $ , and A₀ = −1.

    (2) gcd{i|Ai ≠ 0} = 1.

    (3) f is antipalindromic, i.e., for fpal(x) := x²nf(1/x), we have fpal(x) = −f(x).

    Theorem 24.1. For χ₂ the quadratic character of k×, form the object $ N \, := \, \cal{L}_{\chi_2(f)}(1/2)[1] \, \in \, \cal{P}_{arith} $ . We have the following results.

    (1) Ggeom,N = O(2n).

    (2) Choose a nontrivial...

  28. CHAPTER 25 G₂ Examples: the Overall Strategy
    (pp. 147-154)

    In this and the next two chapters, we fix, for each prime p, a prime $ \ell \, \neq \, p $ and a choice of nontrivial $ \mathbb \overline {Q_\ell}^\times $ -valued additive character ψ of the prime field $ \mathbb{F}_p $ . Given a finite extension field $ k/\mathbb{F}_p $ , we take as nontrivial additive character of k the composition $ \psi_k \, := \, \psi \, \circ \, \text{Tr} _{k/\mathbb{F}_p} $ , whenever a nontrivial additive character of k is (implicitly or explicitly) called for (for instance in the definition of a Kloosterman sheaf, or of a hypergeometric sheaf, on $ \mathbb{G}_m/k $ ). Given a finite field k of characteristic p, and a (possibly trivial) multiplicative character χ of k× which, if...

  29. CHAPTER 26 G₂ Examples: Construction in Characteristic Two
    (pp. 155-162)

    We treat the case of characteristic two separately because it is somewhat simpler than the case of odd characteristic. Recall from the first paragraph of the previous chapter that for k a finite field of characteristic 2, and any character χ of k×, the Tate-twisted Kloosterman sheaf of rank seven

    $ \cal{F}(\chi, \, k) \, := \, Kl(\mathbb{1}, \, \mathbb{1}, \, \mathbb{1}, \, \chi, \, \chi, \, \overline \chi, \, \overline \chi)(3) $

    has Ggeom = Garith = G₂. Our first task is to express its stalk at a fixed point $ a \, \in \, k^\times $ as the finite field Mellin transform of the desired object N(a, k).

    We abbreviate

    $ Kl_2 \, := \, Kl(\mathbb{1}, \, \mathbb{1}), \qquad Kl_3 \, = \, Kl(\mathbb{1}, \, \mathbb{1}, \, \mathbb{1}). $

    Lemma 26.1. For a finite extension $ k/\mathbb{F}_2 $ , and for $ a \, \in \, k^\times $ , consider the lisse perverse sheaf...

  30. CHAPTER 27 G₂ Examples: Construction in Odd Characteristic
    (pp. 163-172)

    The situation in odd characteristic is complicated by the quadratic character χ₂: the ! hypergeometric sheaf of type (7, 1)

    $ \cal{F}(\chi_2, \, k) \, := \, (A^{-7})^{deg} \, \otimes \, \cal{H} (\mathbb{1}, \, \mathbb{1}, \, \mathbb{1}, \, \chi_2, \, \chi_2, \, \chi_2, \, \chi_2; \, \chi_2), $

    though lisse on $ \mathbb{G}_m $ , is not pure of weight zero, nor is its Ggeom the group G₂. One knows [Ka-ESDE, 8.4.7] that this sheaf sits in a short exact sequence

    $ 0 \, \rightarrow \, V \, \otimes \, \cal{L}_{\chi_2} \, \rightarrow \, \cal{F}(\chi_2, \, k) \, \rightarrow \, (A^{-7})^{deg} \, \otimes \, Kl(\mathbb{1}, \, \mathbb{1}, \, \mathbb{1}, \, \chi_2, \, \chi_2, \, \chi_2) (-1) \, \rightarrow \, 0, $

    with

    $ V \, := \, (A^{-7})^{deg} \, \otimes \, H^1_c(\mathbb{G} _m / \overline {k}, \, Kl(\mathbb{1}, \, \mathbb{1}, \, \mathbb{1}, \, \chi_2, \, \chi_2, \, \chi_2)). $

    The rank 6 quotient $ (A^{-7})^{deg} \, \otimes \, Kl(\mathbb{1}, \, \mathbb{1}, \, \mathbb{1}, \, \chi_2, \, \chi_2, \, \chi_2)(-1) $ is pure of weight 0, and the rank one subobject $ V \, \otimes \, \cal{L}_{\chi_2} $ is pure of weight −4; in fact Frobk|V = A⁴, cf. [Ka-GKM, 4.0].

    In order to construct the objects N(a, k), in odd characteristic we will first construct an...

  31. CHAPTER 28 The Situation over $ \mathbb{Z} $ : Results
    (pp. 173-180)

    Suppose we are given an integer monic polynomial $ f(x) \, \in \, \mathbb{Z} [x] $ of degree n ≥ 2 which, over $ \mathbb{C} $ , is “weakly supermorse,” meaning that it has n distinct roots in $ \mathbb{C} $ its derivative f′(x) has n − 1 distinct roots (the critical points) $ \alpha_i \, \in \, \mathbb{C} $ and the n − 1 values f(αi) (the critical values) are all distinct in $ \mathbb{C} $ . Denote by S the set of critical values. Suppose that S is not equal to any nontrivial multiplicative translate aS, for any a ≠ 1 in $ \mathbb{C} ^\times $ . It is standard that for all but finitely many primes p, the reduction...

  32. CHAPTER 29 The Situation over $ \mathbb{Z} $ : Questions
    (pp. 181-186)

    There is another sense in which we might ask about “situations over $ \mathbb{Z} $ ,” namely we might try to mimic the setting of a theorem of Pink [Ka-ESDE, 8.18.2] about how “usual” (geometric) monodromy groups vary in a family. There the situation is that we are given a normal noetherian connected scheme S, a smooth X/S with geometrically connected fibres, and a lisse $ \mathbb \overline {Q_\ell} $ -sheaf $ \cal{F} $ on X of rank n ≥ 1. For each geometric point s in S, we have the restriction $ \cal{F}_s $ of $ \cal{F} $ to the fibre Xs. Pick a geometric point xs in Xs. Then for...

  33. CHAPTER 30 Appendix: Deligne’s Fibre Functor
    (pp. 187-192)

    In this appendix, we prove Theorem 3.1, i.e., we show that $ N \, \rightarrow \, \omega(N) \, := \, H^0(\mathbb{A} ^1 / \overline{k}, \, j_{0!} N) $ is a fibre functor on the Tannakian category $ \cal{P}_{geom} $ of those perverse sheaves on $ \mathbb{G}_m/\overline{k} $ satisfying $ \cal{P} $ , under middle convolution. Throughout this appendix, we work entirely over $ \overline{k} $ , explicit mention of which we will omit. Thus we will write ω(N) simply as $ H^0(\mathbb{A} ^1, \, j_{0!}N) $ . And when we wish to emphasize the roles of both 0 and ∞ in its definition, we will write it as $ \omega(N) \, := \, H^0(\mathbb{P}^1, \, Rj_{\infty_*}j_{0!}N) = H^0 (\mathbb{P}^1, j_{0!} Rj_{\infty_*}N) $ .

    It will be convenient to define, for any object $ M \, \in \, D^b_c (\mathbb{G}_m, \, \mathbb \overline{Q_\ell}) $ ,

    $ \omega(M) \, := \, H^*(\mathbb{P}^1, j_{0!} \, Rj_{\infty_*}M) \, := \, \oplus_{i\in\mathbb{Z}} H^i(\mathbb{P}^1, j_{0!} \, Rj_{\infty_*}M). $

    Lemma 30.1. If M is perverse, then...

  34. Bibliography
    (pp. 193-196)
  35. Back Matter
    (pp. 197-203)