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SUBJECT INDEX

Abel map 48

Abelian integral 64

Abhyankar, S. 97

Absolute values (conventions) 20

A-disc, A-codisc 224

Adelic set 322,336

Adelic neighborhood 322,336

Ahlfors, L. 160,162

Algebraic capacitability 15,258

PL-domains are A.C. 258

RL-domains are A.C. 260

compact sets are A.C. 262

finite unions, intersections of

RL-domains, compact sets

are A.C. 266

finite unions of open balls,

closed balls are A.C. 274

pUllbacks of A.C. sets are A.C.

274,304-309

non-A.C. sets 263-264

Choquet's theorem fails 265

not stable under intersections

266

A.C. sets are closed 276

A.C. sets are approximatable by

PL-domains, RL-domains

311-316

259

pathological examples 264-266

Algebraic curves (conventions) 21

Algebraic integers 1,9,373,376

Algorithm

for finding minimal model 101

for finding val{r) 367

Analytic arc 162,165,167

Approximation of

(Archimedean case):

basic theorem 168

input to Fekete-Szego Theorem

171,173

independent variability of

leading coeffs 173-184

(Nonarchimedean case)

input to Fekete-Szego Theorem

318,319

Arakelov function «x,yD v

13,14,80,89

(Archimedean case);

axioms for Arakelov Green's

functions 80

non-normalized 80

for p1 80

for elliptic curves 82

for curves of genus g 2 83

in terms of 86

(Nonarchimedean case):

definition 89

for p1 90

for curves w/good reduction 90

for Tate curves 93

for curves of genus g 2

96,115-116

comparability with Ux,yU v 128

Artin, M. 97,118

Artin Contraction Theorem 19

Balancing 397

Ball

conventions 22,30,185

boundary of 30

parametrizable 30

isometrically parametrizable

31,185-186

rational, irrational 186

Barrier 143

Basis functions 382

Beilinson, A.A. 18

Bent line segments 342

Bertrandias, F. 2

Blowing up 101,102,117,118,121

Boundary

of baIlor disc 30

of RL-domain 50

of R-disc 239

of disc or codisc 239

of island domain vD 239

Borel measure 135,186

Bourbaki, N. 45,321

Brown, S. 30

e(KA) 322,336

Canonical distance 12,14

defini tion 57

factorization property 57

Galois stability 57,61

construction of 58,63

characterization of 69

approximatable by (l/N)loglf(z) I

72

change of centers formula 72

Gross's formula for 13,77,106

invariance under base change 90

430

joint continuity in x,y

57,61,63,66,67,69-70

in terms of pairing

77,106

weak triangle inequality

73,127

(Nonarchimedean case):

for pl 90

for curves w/good reduction 90

for Tate curves 93

weak triangle inequality 127

local ultrametric inequality

130

Canonical metric

103,221

comparability with Hx,Yftv 120

ultrametric inequality for

104,224

Cantor, D. 2,6,10,16,50,220,226,

320,328,332,355,369,381

Cantor set 347

Capacity

(Archimedean case)

sets of capacity 0

133,134,137,137

definition:

for compact sets 136

for arbitrary sets 137

capacity of E* 146

unchanged by sets of capacity

capacity 0 149

limits of open sets 149

equality with

151

examples 338-347

table of capacities 348-351

431

(Nonarch1medean case)

definition:

r1;(E) for compact sets 190

r1;(U) for PL-domains 259

r1;(F) for general sets 259

countable unions 193

inner capacity 15,192,259

unchanged by capacity 0 sets 199

outer capacity r1;(E) 15,259

for countable sets 200

limits of open sets 200

unions of compact sets 201

equality with d1;(E), q1;(E) 204

examples 211-219,352-365

for Elliptic curves 360-365

algorithm for computing capacity

of a union of sets 353-355

dependence on ground field 352

(Global capac1ty) 3,10,171,163

definition of r(F,I) 328

monotonicity properties 331-332

separation inequality on pI 333

base extension 333

puLlbacks 333

algorithm for computing val(f)

366-367

for elliptic curves 5,370

Carrier 140,146,195

Cauchy estimates 34,35,36

Centers Xi £ I 322

Chebyshev constants

q1;(E) 7,150,203

150,203

203

Chinbarg, T. 19,79,103

Choqaet, G. 265

Chordal distance 23,25,26

as Arakelov Green's function

81

Circuit model 110

Codisc 22

Conductor potential u E(z,1;)

(Archimedean case):

definition 137

continuity properties 142

constant on E 143

uniqueness 145

(Nonarch1medean case)

definition 191

constant on E 195

less than V1;(E) off E 197

uniqueness 211

examples 211-219

for rings of integers 212

for pullbacks 214

for finite unions 216

pathological examples 217

Continuum 156

Convex hull 174

Coordinate patches 133

CPA(£(e)v) 113,115

Currents 86

aD!; ("outer boundary") 143,146

Ax ("sum of slopes" operator)

113,256

deg(f) 21

Degree sequence 381

Deligne-Mumford Theorem 101,118

Dictionary between circuits,

harmonic functions on graphs

110

432

Differentials of 1st, 3rd kinds 64

Dirichlet-Minkowski Unit Theorem

395

Disc

conventions 22,356

boundary of 30

open, closed 30

isometrically parametrizable 30

with respect to "x,y"tt,v 222

Divisor function 113

Dwork, B. 17,39,40

140,146,195

Electric circuit analogy 110

Electrostatic analogy

for capacity 6,8

Elliptic curve 5,361,370

Energy integral 136

Equilibrium distribution 7

(Archimedean case):

definition 137

uniqueness 145

formula in terms of G(z,oo;E) 162

examples of 163

(Nonarchimedean case):

defini tion 190

uniqueness 211

examples 211-219

for rings of integers 212-213

for pullbacks 214

for finite unions of sets 216

pathological examples 217

Evans function 208

Exponential map 49

Fekete,M. and Szego,G.

1,2,3,4,6,9,15,373

Fekete-Szego Theorem

in classical case 1,2,373,376

for algebraic curves 4,414,415

when r(F,<) = 1 416

archimedean input 171,173

nonarchimedean input

316,318,319

inner vs outer capacity

appropriate for 418-420

Fekete points 152

Fine cover 187

Fine subcover 187

Fresnel, J. 220

Frobenius's Theorem 328-329

Frostman's Theorem 140,195,284

Fubini-Study metric 23,26

FUlton, W. 117

Function suitable for

defining 59

Fundamental Group 63

Fundamental Theorem of Game

Theory 327,335

Gal(K/K): action on

probabili ty vectors 321

Generalized disc 222

Generic value of I f(z) I v 223

ga(r,f) 224

224

Geodesic 26

Gillet, H. 18

Global mapping functions 381

when r(F,<) < 1 384-393

when r(F,<) > 1 395-413

433

Good reduction 91,322,323,383,411

Green's function

8,10,14,155,277

(Archlmedean case):

definition of 155

basic properties 156

positivity off E 156

monotonicity 157

behavior under limits 158-159

symmetry 160

continuity in two variables 161

functoriality 163,164

approximatable by

of good sets 165

approximatable by (l/N)loglf(z)1

168

(Nonarchimedean case):

upper: 208

upper: 277,282

lower: 277,282

well defined 278

278,282

examples of non-A.C. sets

where upper, lower Green's

functions differ 282-283

= off F,

for A.C. sets 283,291,293

examples with

on F 295

sequence Green's functions

283

the zero set 292

definition:

for A.C. sets 297

monotonicity in F 297

symmetry of 299

continuity of 301

behavior under pullbacks 305

Galois stability 310

dependence on ground field 352

Green's identity 67,87,160,161

Green's matrix

local Green's matrix f v

11,324,326

f v = 0 for almost all v 324

global Green's matrix f(F,X)

1,325,326

f(F,l) well-defined 327

reducible vs irreducible

328,369

Gross, B. 12,19,63,74,77,83,

89,106

Grothendieck, A. 99

Gunning, R. 63,64,65

Haar measure 23

Hadamard Quotient Theorem 18

Half-disc 340

Harbater, D. 19,27

Harmonic function 63,73

Harnack's principle 70

Hartogs, F. 65

High order coefficients

374,378,389,408

Hilbert's Lemniscate Theorem 168

Hilbert's 10th problem for 0 6

Hille, E. 168,374

Hole (of codisc) 223,225

Horizontal divisor 98

Hri1jac, P. 106

Hyperbolic polygon group 83

434

Independent variability of leading

coefficients 173

Inner capacity 192

for a PL-domain 258

Intersection pairing 98

on special fibre 98

for horizontal divisors 99

for divisors of functions 100

Intersection theory 97

on regular surfaces 97

on semi-stable models 101

on well-adjusted models 103

formula for pairing 106

iv(x,y)

for Tate curves 94-96

for curves of arbitrary genus

96,107

96,107

Island domain 220

definition 222

boundary vD 239

equivalence with PL-domains

239,244,252

RL-domains = intersection of 224

Isometric parametrization

31,32,42,46,54,62,185

Jacobian Construction Principle 48

Jacobian matrix 31,32,37

jv(x,y)

for Tate curves 94-96

for curves of any genus 116

jz(X,y}, 107,109,110,222

mean value property for 108

satisfies Laplace equation 109

extension to e(Kv} 120

continuity of 112

takes rational values

at rational points of R(e)v

112

polarization identity for 112

Kakeya, S. 381

Kani, E. 19,74,86

Karlin, S. 174,327

Kirchoff's laws 110

KA 336

Kodaira symbol 361,364

K-symmetric

set c elK) 321

probability vector, matrix 321

set F c e(KA) 322

Kv-symmetric set F c e({lv)

311,321

Leading coefficients 386,403

Lichtenbaam, S. 90,97,101

Linear programming 367

Line segment 340

Lipman, J. 97

Local-global principle 6

Logarithmic leading coeffs 384

independent variability of

173,399

Logarithmic potential function

135,186

Logarithmic singularities 67

logv(x} 20,185

Low order coefficients 386,392

Lower Green's function

see Greens function

Lower semi-continuous function

134

Manin, Y. 93

Maria's Theorem 139,191

Matignon, M. 220

Matrix Y, nonsingularity of 254

Maximum Principle

for divisor functions 114

for harmonic functions 67

for power series 45

for algebraic functions

and RL-domains 51,59,60

435

Nllron model 94

Newton's method for power series

30,31

Newton polygon 40,41,52,53

Normalized logarithm

(l/N)loglf(z)1 165,188

Outer boundary 143,146

Outer capacity 259

Metric Patching argument for Fekete,

Fekete-SzegB Theoremschordal 23,25,26

Fubini-Study 23,26

spherical Rx,yH v 24,56

Ix,yRtt,v 103,120,221,224

on pN 122-127

Minimal model 101

in classical theorems

in adelic theorems

(case r(lF,l) < 1 )

(case r(IF,l) > 1 )

PL-domain 9,10,14

375,378

384-393

395-413

algorithm involving blowups 101

Minimax theorem 327

Mittendorf, W. 112

Mollifier 93

Moret-Bailly, L. 17,19

Multinomial Theorem 378,408

Mumford, D. 26

n-armed star 343

Negative definiteness 331

Nllron, A. 2,23,25,74,94,101

Nllron's local height pairing

2,12,13,74

normalization of 74

axioms for 74-75

extension to ZO(e(Kv » 76

Gross's formula for 77

via Arakelov functions 75

intersection theory formula 106

def ini tion 236

equivalence with island domains

239,244,252

finite intersections of 251

definition 50,236

independent of 244,252

finite unions of isometrically

parametrizable balls 252

Poincarll series 83

Poisson kernel 71

P6lya-Carlson Theorem 17

Potential function

135,186

(see also Conductor potential)

approximation by (1/N)loglf(z)1

169,188

lower semicontinuity 135,194

superharmonicity 135

436

Prime function 63,64

Probability measure 135,186,190

Probability vector 171,311,321,383

action of Gal(Kv/K) on 321

Product formula 325,415

Proper map 48

Pullback formula

for capacity 1,2,333,344

for Green's functions 163,305

Punctured disc 220

Punctured A-disc 224,226

PV-numbers 17

Quadraticityargument 140,141,195

qv 20

222

Rational points on reduction graph

104,117,221

Rearrangement Lemma 40,47

Reduction graph 96,104,221

construction of 104

natural metric on 104

"rational points" on 104

Reduction map 105,117

definition on e(Kv ) 105

extension to e(Kv ) 117

Regular n-gon 341

Resistor model 110

Riesz Decomposition Theorem 144

Riemann-Roch Theorem 175,382

RL-component

of complement of RL-domain 236

of complement of A.C. set 304

RL-domain

definition 14,50,220

boundary 50

finite unions of balls are 54

as intersection of island

domains 224,252

unions, intersections 251

Robbins's constant 136,190

Robinson, R. 16,369

Roquette, P. 6,17

Rothstein, W. 65

Sa({E j } ) , Sa({D j } ) 285

292,293

Salem, R. 18

Saddle point 367

Semi-stable reduction 96,101

Separably algebraic extension 21

Separation inequality 332

Serre, J.P. 49,59

Sbatarevlcb, l.R. 99,101

Shimura, G. 91,322,325,383

Soulfil, C. 18

Spherical metric 24,56

Strong Approximation Theorem 389

S-integer 391

S-unit 386,403

S-subunit 387

SUO, 1) 83

Subharmonic function 135

Superharmonic function 135

Symmetrization bounds 345

Szplro, L. 17

Table of capacities 348-351

Tangent discs 341

Tate curve 93,360

Tonelli's Theorem 147,198,212

437

Transfinite diameter 150,202

Triangle inequality

for metric on pN 122

for (weak global) 129

for (strong local) 130

Tsuji, M. 19,135

Two Options Theorem 328

135

Uniformizing parameter

determining 57,188

Universal covering space 63

Upper Green's function

see Green's function

Upper semi-continuous function 134

v-component 221

Value of f as a matrix game

definition of val(f) 327

has same sign as largest

eigenvalue of f 328

determininant criterion for

negative definiteness 331

algorithm for computing 365-366

belongs to Q·log(p) in

characteristic p 367,397

van der Poorten, A.J. 18

Varley, R. 19,63,89

rOt.t(w) 107

Vertical divisor 98

("Robbin's constant")

definition 136,190

extremal properties 136,147,197

bound for unions of sets 148,198

Weak convergence 137,190

Well, A. 27,28,38,382

Weil distribution 27

Weil reciprocity 59,60,300

Weierstrass points 382

Weierstrass Preparation Theorem

43

Welssaur, R. 19,63

Well-adjusted model 103

«-capacitable set 327,414

«-trivial set 323

(l,a)-function 383