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REFERENCES
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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