Index

ψv-regular sequence, see also regular

sequence

wv, see also distinguished place wv

Abel map, 179, 181, 187, 391, 392, 409,

411, 417

adele ring, 193

aﬃnoid

admissible, 398

aﬃnoid domain, x, xviii

Berkovich aﬃnoid, 7, 123, 128

domain, 398, 401

strict closed aﬃnoid, 128–130

Akhiezer, Naum, 14

algebraic integer, xii, xiv, 27, 28, 30, 31, 33

totally real, ix, xii

algebraically capacitable, xviii, xix, 1, 2, 4,

5, 6, 20, 28, 39, 80, 91, 106, 114, 117,

119, 120, 125, 129, 131, 159

with respect to ζ, 80

algorithm to compute nonarchimedean

capacities, 22

analytically accessible, 89, 104–107, 116

apportionment in House of Representatives,

160

Arakelov functions, 74

Arakelov theory, xiii

arc, 334, 385

analytic, 378

circular, 58

smooth, 104, 105, 135

Argument Principle, 11, 261

Autissier, Pascal, xiii

Baker, Matthew, 5, 336

Balinski, Michael, 160

band

BandN (k), 214, 215, 219, 221, 233, 234,

237, 238, 239, 249, 250, 257, 259, 264,

269–272, 274, 280–284, 295, 296, 298,

301, 312, 313, 315–317

coeﬃcients patched by bands, 191, 214,

216, 221, 231, 234, 237

simultaneously when char(K) = p 0,

231, 233–235

high-order, 295

barrier, 334

Basic Patching Lemma, 287, 291, 292, 309

basic well-distributed sequence, 98, 171

basis

L-rational, xv, 61, 64, 65, 66–69, 191,

197, 209, 211–213, 219, 224, 230–232,

234, 243, 244, 247–249, 251, 257, 262,

263, 269, 271, 275, 280, 282, 299–301,

326–329

Lsep-rational,

64, 65, 66, 67–69, 197,

198, 224, 230, 231, 234, 235, 237, 238,

241, 243, 247, 248, 269, 280, 328

scaled L-rational, 242, 243, 246

Berkovich

adelic neighborhood, 6, 129

adelic set, 6, 7, 128, 129

analytic space, 5, 120, 336

analytification, 120, 125

closure, 121, 125, 128

compact set, 130

curve, xii, xix, 120, 122

Green’s function, 120, 125

neighborhood, 7, 128

open set, 7, 130

quasi-neighborhood, 6, 7, 128–130

strict closed aﬃnoid, 7, 128, 130

topology, 6

Berkovich, Vladimir, 5, 120

Bertrandias, Fran¸ coise, xii

binomial theorem, 251

block, Galois

Block(i, j), 214, 220, 237, 238

coeﬃcients patched block by block, 214,

219, 220

boundary, 111

equilibrium distribution supported on, 37

exceptional set contained in, 81

of filled Julia set, 19

piecewise smooth, xiii, 103, 105, 133,

135, 202, 249, 257, 261

Brouwer Fixed Point theorem, xv, xxi, 154

427