Index

adapted to Γ, 105–6, 107

additive characters

background on sums involving, 1–4

in Evans and Rudnick examples, 7, 8, 68–70, 71

G₂ examples and, 147

GL(n) examples and, 83–84, 115, 121

O(2n) example and, 145–46

SL(n) examples and, 125–32, 135

additive version, Tannakian

approach to, 16–17

adjunction map, 190–92

antipalindromic polynomials, 145–46, 185

arithmetically self-dual object, 49–50, 51

Artin-Schreier reduced polynomial, 100–101, 125–26, 130, 135

Artin-Schreier sheaf, 8

“adapted to Γ” and, 105

autodual construction and, 59

direct sum of N_i and, 115, 121

Fourier Transform and, 17

GL(n) and, 84

Artin-Schreier-Witt sheaf, 84

autoduality

arithmetically self-dual object, 49–50, 51

first construction of objects, 53–54

geometrically self-dual object, 51

Goursat-Kolchin-Ribet type results, 63–65

of N(a, k), 158, 164, 169–70

second construction in nonsplit case, 61–62

second construction in split case, 55–60

sign of, 49–51

SL(2) and, 59, 68, 70, 73–75.

See also orthogonal autoduality; symplectic autoduality

bad (for N) multiplicative characters, 13, 17–18

bound on the number of, 178

for good p, 173, 174

Legendre sheaf and, 122

in situation of main theorem, 43–44

totally wildly ramified objects and, 18, 101, 135

weights and, 79–80

Carlitz isomorphism, 159

character sums over finite fields

background on, 1–4

Gauss sums, 1, 4, 115, 128–29, 145–46, 147

Kloosterman sums, 1, 2–3, 4

numerical experiments on, 7

See also Evans sums; Rudnick sums

cohomology, étale, 8

cohomology groups, 20, 26–28, 29, 42, 77–78, 117, 132, 159–60, 176

conjugacy classes

equidistribution of, 13–15, 40–43, 49–50, 147–50, 173–75.

See also Frobenius conjugacy classes

connectedness, 46–47

convolution

Mellin transform and, 9

of perverse sheaves, 19–20

See also middle convolution

!-convolution, 9–10, 19

middle convolution and, 10, 19, 114, 121, 187

product formulas for, 10

cubic polynomials, squarefree, 2–3

cyclic groups, 37, 39–40, 45–46

cyclotomic field, 34, 129

Deligne’s equidistribution theorem, 3

additive case and, 17

Deligne’s fibre functor. See fibre functors

Deligne’s main theorem in Weil II, 152, 170

Deligne’s semicontinuity theorem, 106, 157, 165

delta objects, 35–36, 63, 79–80, 141, 142, 182

det (N). See Tannakian determinant

“dimension” of perverse sheaves

Euler characteristic as, 10

Euler-Poincaré formula and, 67, 68, 70, 74, 75

generic rank and, 178

normal noetherian connected scheme and, 181, 182

number of bad characters and, 178

direct sum of objects N

GL(n) and, 65, 113–24

Goursat-Kolchin-Ribet type results, 63–65

orthogonal examples and, 104–5

SL(2) and, 70, 71, 74

SL(n) and, 131–32, 143

Sp(n) and, 90, 93, 96, 98–102

drop, 67

eigenvalues. See Frobenius eigenvalues

elliptic curves

character sums and, 2–3

Legendre family of, 73–75

equation vectors, 79

equidistribution

in additive case, 15–16

of conjugacy classes, 13–15, 40–43, 49–50, 147–50, 173–75

Deligne’s theorem, 3, 17

introduction to questions about, 3–4

main theorem on, 13–15, 39–44

numerical experiments on, 7 (see also Evans sums; Rudnick sums)

of r-tuples of angles of Gauss sums, 115

of r-tuples of angles of Jacobi sums, 115

Sato-Tate conjectures on, 3 of sums S, 10–11

étale cohomology, 8

Euler characteristic, 10, 20, 25, 67, 166, 182

Euler-Poincaré formula, 67, 68, 70, 74, 75, 152, 177

Evans sums, 7, 8, 10, 16, 68–69, 101, 102

exceptional groups, 16. See also G₂

external tensor product, 9

fibre functors, 11–12, 15–16, 21–24, 187–92

finite extension field and, 29, 31

finite field and, 26–28

G_arith,N and, 11–12, 33–34, 77–79

n’th power homomorphism and, 45

“forget supports” maps, 12, 19, 21–23, 26–29, 187–92

Fourier Transform, 16–17

fraction field Q_ℓ, 8

Frobenius automorphism, 11

fibre functors and, 26–27, 29, 31, 77

Frobenius conjugacy classes, 12, 31–32

G₂ and, 147–48, 149–50, 151–52

with power having all equal eigenvalues, 35–36

quasiunipotent, 33–34, 36

quotient group and, 39

semisimplified, 13, 18, 49, 147–48

Tannakian determinant and, 107, 108, 125–28, 141.

See also traces of Frobenii

Frobenius eigenvalues

absolute values of, 18, 122

all equal, 35–36

all roots of unity, 33–34, 36

Frobenius torus and, 79

unitary, 13, 31–32

weights of, 77–78, 159–60, 170

Frobenius element, 11–12

Frobenius-Schur indicator, 49

Frobenius tori, 18, 77–79

orthogonally self-dual objects and, 103–4

SL(n) and, 90, 142–43

symplectically self-dual objects and, 89–90, 98, 99

G₂, 16

obtaining in characteristic two, 155–62

obtaining in odd characteristic, 163–71

overall strategy, 147–54

G_arith,N and G_geom,N

equality of, 13–15, 16, 43, 178

fibre functors and, 11–12, 33–34, 77–79

Frobenius conjugacy classes and, 31

getting elements of G_arith,N, 77–80

GL(n) examples and, 81–86

group-theoretic facts about, 33–38

Legendre family and, 73–75

normal noetherian connected scheme and, 182–83

notation for, 4

orthogonal examples, 103–11

quotient groups formed from, 37, 39–40, 45–46

as reductive groups, 13, 32, 33, 39, 43

symplectic examples and, 89–102

Gabber-Loeser results, 15, 181–82, 187

Gabber’s theorem on

prime-dimensional

representations, 130, 132, 138, 149, 161

Gauss sums, 1, 4, 115

quadratic, 128–29, 145–46, 147

generic rank

“bad for N” characters and, 13

bound on, 174–75, 178

condition P on perverse sheaf and, 10

drop and, 67

general inequality for, 175, 176–78

main theorem and, 14, 15

of middle convolution, 116, 135–36

nonpunctual one-dimensional object and, 116–20, 121, 122

geometrically self-dual object, 51

GL(2), 59

GL(n)

applying Theorem 28.1 on general

G to, 178

direct sum of objects N and, 65, 113–24

examples involving, 16, 81–86

normal noetherian connected scheme and, 181–83

for situation over Z, 173

GL(n – 1), 85–86, 173

GL(n) × GL(n) × … ×

GL(n), 113–24

GL(!(N)), 11

GL(!₀(N)), 13, 16

good for f, of prime p, 173

good (for N) multiplicative characters, 12–15, 21–23

autoduality and, 49

Frobenius conjugacy classes and, 31–32, 33

main theorem and, 40

structural group and, 29

Tannakian determinant and, 141

Goursat-Kolchin-Ribet theorem, 63–65, 104, 115, 130, 132, 143

Goursat’s lemma, 161, 184–85

Haar measure, 4, 7n, 14, 40, 43

Hasse-Arf theorem, 157

Hasse’s theorem, 73, 74

hypergeometric sheaves

bad characters and, 79–80

G₂ examples and, 147, 163, 164, 165

local monodromies of, 80, 83, 92–93, 94, 108

middle convolution of, 35, 63, 114

orthogonally self-dual, 99–100

symplectically self-dual, 108, 110–11

Tannakian determinant and, 35, 82–86, 113–15, 122–23, 124

inertia groups

Euler-Poincaré formula and, 67

Kummer sheaf and, 22

notation for representations of, 4–5

tame, 12

totally wildly ramified

representations of, 136–38

isogenies, 45–47

Jacobi sums, 115

Kloosterman sheaves, 122, 130–32

arithmetic determinant formula for, 132

G₂ examples and, 147, 155, 159–60, 163–64

Tate-twisted of rank seven, 147, 155, 156

Kloosterman sums, 1, 2–3, 4

Kowalski, Emanuel, 4, 173

Kummer sheaves, 8

condition P on N and, 10, 19–20

fibre functors and, 21–23

“good for N” characters and, 12–13, 29

SL(2) and, 69

symplectic object and, 102

Lang torsor, 29

Laurent polynomial of bidegree

(a, b), 125–30, 135, 138, 140

Lefschetz Trace formula, 10, 42, 53, 95, 132, 152

Legendre family of elliptic curves, 73–75

Legendre sheaf, 82–83, 91–96, 109–10, 122

Leray spectral sequence, 189–91

Lie-irreducibility, 46–47

GL(n) examples and, 81, 86

Legendre family and, 73–75

of N(a, k), 160–62, 171

SL(2) examples and, 68–69

lisse sheaves adapted to Γ, 105–6

condition P on N and, 10

middle convolution of, 106

monodromy groups attached to, 4

notation for, 4

pure of some weight, 8

local monodromies

“adapted to Γ” and, 105

in additive case, 17

bad (for N) multiplicative characters and, 17–18, 32

of hypergeometric sheaves, 80, 83, 92–93, 94, 108

of orthogonally self-dual objects, 100, 103, 104

of symplectically self-dual objects, 89, 90, 91, 94, 98–99, 101

tame characters in, 12–13

tame or tame part of, 90, 104–6

unipotent, 109–10, 165, 171

weakly supermorse polynomials and, 85–86.

See also monodromy groups

main theorem, 13–15, 39–44

Mellin inversion, 159

Mellin transform, 9, 155. See also traces of Frobenii

middle convolution, 10, 19, 20

adapted to Γ, 105–6

additive, 17

autoduality and, 51, 63, 65

!-convolution and, 10, 19, 114, 121, 187

fibre functor and, 187

generic rank of, 116, 135–36

of hypergeometric sheaves, 35, 63, 114

n’th power homomorphism and, 45

punctual objects and, 35–36, 38

of semisimple objects in P, 23

subcategory of P with stability under, 25–26

of totally wildly ramified objects, 135–36

monodromy groups

additive case and, 17

Legendre family and, 73–74, 91

normal noetherian connected scheme and, 181

notation for, 4

symplectic, 108

weakly supermorse polynomials and, 85

Zariski closure of, 17.

See also local monodromies

multiplicative characters, 7–8

in character sums, 1–4.

See also bad (for N) multiplicative characters; good (for N)

multiplicative characters;

Kummer sheaves; quadratic characters

N(a, k)

construction in characteristic two, 155–62

construction in odd characteristic, 163–71

as self-dual object, 158

Newton symmetric functions, 34, 127

noetherian scheme, normal connected, 181–83

O(n), 103–5, 111, 178

O(2n), 16, 145–46, 178

orthogonal autoduality, 103, 107–11, 145–46, 147–49, 151–52, 158–61, 171. See also autoduality

orthogonal similitudes, 185

P. See perverse sheaves satisfying P palindromic polynomials, 96–98, 99, 142, 143–44, 183–84

perverse sheaves

in additive case, 16–17

convolution of, 19–20. See also middle convolution

direct sum of. See direct sum of objects N

Fourier Transform of, 16

Kloosterman sheaf constructions of, 155–58, 163–64

negligible, 20, 25, 166, 167, 169, 187

notation for, 4

P_arith subcategory of, defined, 25

pure of weight w, 8

pure of weight zero, 12, 13

Tannakian categories of, 10–11, 20, 25, 158.

See also “dimension” of perverse sheaves

perverse sheaves satisfying P, 10–11

additive version of P, 17

arith and geom subcategories of, 11

convolution and, 19–20

Deligne’s fibre functor on, 21–24.

See also fibre functors

main theorem on, 13–15

normal noetherian connected scheme and, 181, 183

semisimple object of, 17, 20, 23

Peter-Weyl theorem, 13, 14, 40, 148, 153

polynomials

antipalindromic, 145–46, 185

Artin-Schreier reduced, 100–101, 125–26, 130, 135

character sums involving, 1–3

of degree n, not a power of p, 141

GL(n) and, 84–85, 123–24

Laurent, of bidegree (a, b), 125–30, 135, 138, 140

noetherian schemes and, 183–85

O(2n) example and, 145–46

palindromic, 96–98, 99, 142, 143–44, 183–84

squarefree cubic, 2–3

symplectically self-dual objects and, 96–99, 102

weakly supermorse, 85–88, 173, 178

prime fields

character sums over, 1–4

equidistribution results over, 4, 174

number of squares in, 1

punctual objects, 15, 33–38

pure of weight w, 8

pure of weight zero, 8, 11, 12, 13, 15, 16–17

quadratic characters

character sums involving, 1–3

N(a, k) construction and, 163

O(2n) and, 145–46

palindromic polynomial and, 99

SO(n) and, 108, 110

quadratic extension, 25, 61, 81, 89, 103, 111, 179

quadratic Gauss sum, 128–29, 145–46, 147

quasiunipotent Frobenius conjugacy classes, 33–34, 36

rank. See generic rank

Riemann Hypothesis for curves over finite fields, 2

Rudnick sums, 7, 8, 10, 16, 69–71, 102

Sato-Tate conjectures, 3. See also equidistribution

Sato-Tate measure, 3, 7, 16

self-dual object. See autoduality

semicircle measure, 3

semisimple objects in P, 17, 20, 23

skyscraper sheaf, 10, 15

SL(2), 16, 59, 67–71, 73–75

SL(d), 65

SL(2g), 185

SL(n), 16

applying Theorem 28.1 on general G to, 178

direct sum of N_i and, 113, 115

examples for n an odd prime, 125, 129–33

examples for n not a power of p, 141–44

examples with slightly composite n, 135–40

Frobenius tori and, 90, 142–43

SO(3), 74

SO(7), 147, 149–50

SO(d), 64–65

SO(n), 16, 103–5, 107–8, 110–11

applying Theorem 28.1 on general G to, 178–79

Frobenius tori and, 90

Sophie Germain primes, 139, 140

Sp(d), 55, 57, 59, 62, 63–64

Sp(2g)

maximal torus of, 184, 185

noetherian schemes and, 183–85

Sp(n), 16, 89–102

applying Theorem 28.1 on general G to, 178

Sp(n + 1), 94–96

Sp(2n), 16

squarefree cubic polynomials, 2–3

SU(2), 16

measure on, by trace map, 7n

Swan conductor

in Euler-Poincaré formula, 67

GL(n) and, 121

symplectic autoduality examples of, 89–102

normal noetherian connected scheme and, 183

open problem on, 100–101

SL(2) and, 68, 70, 71

SO(n) and, 108, 109.

See also autoduality

symplectic similitudes, 184

tame characters, 12–13

tame fundamental group, 12, 20, 29, 43

tame inertia group, 12

tame local monodromy, 90, 104–6

Tannakian categories of perverse sheaves

objects satisfying P, 10–11

P_arith subcategory of P, 25, 158

Perv/Neg quotient category, 20.

See also perverse sheaves satisfying P

Tannakian determinant

direct sum of N_i and, 113–15, 121, 123–24

Frobenius conjugacy classes and, 107, 108, 125–28, 141

GL(n) examples and, 82–83, 84, 85, 86

Laurent polynomial of bidegree (a, b) and, 125–30

in orthogonal case, 104–5, 107, 108, 111

and polynomial of degree n, not a power of p, 141, 142

punctual objects and, 35, 37, 82

Tannakian duals, 38, 121, 122, 123, 125, 130, 133, 142

Tannakian groups. See G_arith,N and G_geom,N.

Tate twists

Kloosterman sheaf of rank seven and, 147, 155, 156

notation for, 4–5

traces of Frobenii and, 9, 54

tensor product

in additive case, 17

external, 9

middle convolution as, 10

total drop, 67

totally wildly ramified objects, 18, 101, 135–38

trace map, 7, 15, 43, 61

traces of Frobenii, 8–9, 13, 15

in additive case, 16–17

autodual objects and, 49–51, 53–54, 56–58

bounds on, 15, 151–52

main theorem and, 42–44

Mellin transform and, 9

punctual objects and, 34

SL(a + b) example and, 131–32

symplectic example and, 95

weakly supermorse polynomials and, 86–87

twin primes, 139, 140

UG₂, 147–48, 150, 152–53, 154

U(n – 1), 86, 173

unipotent local monodromy, 109–10, 165, 171

Urysohn’s Lemma, 152–53

variant Sophie Germain primes, 139, 140

vector space subquotients, 11

Verdier dual, 10, 26, 49

adapted to Γ, 106

of N(a, k), 156, 158–59, 164, 169–70

on normal noetherian connected scheme, 181

weakly supermorse polynomials, 85–88, 173, 178

Witt vector, 84

Z

l-adic completion of, 8

questions about situation over, 181–85

results about situation over, 173–79