Journal of Number Theory, v. 104

Introduction......Page 1
Preliminaries......Page 2
The main theorem......Page 3
References......Page 13
Introduction......Page 14
p-adic fields and finite commutative chain rings......Page 16
Preparatory results about p-adic fields......Page 19
References......Page 60
Introduction......Page 62
The hyperbolic classes contribution......Page 65
The identity class contribution......Page 71
The parabolic classes contribution......Page 72
Acknowledgements......Page 73
References......Page 74
Introduction......Page 75
Background and notation......Page 77
A description of the new reduction types......Page 80
New types with residue characteristic 2......Page 81
Valuations of Weierstrass coefficients......Page 83
Charne2,3, general form......Page 84
The families In and Inast......Page 85
Char 3 residue field......Page 86
Char 2 residue field......Page 87
Char 2Kn detail......Page 88
Translating the Weierstrass equation......Page 89
Char(kappa)=2 main procedure......Page 90
Procedure for Kn family......Page 92
Constructing regular models with blow-ups......Page 93
Char 2 Y2 type......Page 94
Conclusions and further work......Page 95
Group schemes and NÕron models......Page 96
Applications and further work......Page 97
References......Page 98
Introduction......Page 100
Siegel modular forms and Siegel-Eisenstein series......Page 101
A question on p-adic Siegel-Eisenstein series......Page 105
Main result......Page 108
Local densities......Page 109
Proof of main theorem......Page 112
References......Page 116
Introduction......Page 118
First bound for L(s,chi)......Page 121
Proof of Theorem 1......Page 124
Some explicit special cases......Page 127
Better asymptotic bounds......Page 129
References......Page 130
Irrationality results for values of generalized Tschakaloff series II......Page 132
Introduction......Page 133
Main proposition (1)......Page 137
Outline of the proof......Page 138
Construction of auxiliary functions......Page 140
Estimation of sigma(r,h)......Page 142
Conclusion of the proof of Theorem 1......Page 144
Proof of Theorem 2......Page 145
Main proposition (2)......Page 147
Construction of simultaneous approximations......Page 148
Nonvanishing of the determinant......Page 150
Proof of Corollary......Page 153
References......Page 154
Introduction......Page 156
Some lemmas......Page 158
Proof of the theorem......Page 160
References......Page 161
Motivation and background......Page 162
Notation and summary of results......Page 163
Proof of Theorems 1 and 2......Page 164
References......Page 169
Introduction......Page 170
Proof of (1) rArr (2)......Page 172
Proof of (2) rArr (3)......Page 173
References......Page 175
Introduction......Page 177
Construction of test functions......Page 179
Kuznietsov trace formula......Page 181
Evaluation of Integrals......Page 184
Weighted distribution......Page 189
References......Page 192
The problem......Page 193
The group S(a,K)......Page 194
The group H1(G,E[a])......Page 197
The rank 2 case......Page 198
Examples of non-trivial S(a,K)......Page 205
References......Page 208
Introduction and Liouville’s type inequalities on a curve......Page 210
Arithmetical applications......Page 216
Diophantine equations......Page 217
Geometrical interpretation......Page 219
Derivations on a curve......Page 222
Around Eisenstein’s Theorem......Page 226
Proofs of Liouville’s type inequalities......Page 230
Proof of Theorem 1......Page 232
Proof of Theorem 2......Page 238
Proof of Theorem 3......Page 245
Proof of Theorem 4......Page 248
Proof of Theorem 5......Page 250
Proof of Theorem 6......Page 252
References......Page 253
When subset-sums do not cover all the residues modulo p......Page 255
Introduction......Page 256
Arithmetic progressions in subset sums......Page 257
Proof of Theorem 1......Page 259
References......Page 262
Introduction......Page 263
Results from the geometry of numbers......Page 265
The cubians......Page 267
Cohn’s theorem on sums of four squares......Page 273
The Icosians......Page 274
Further directions......Page 276
References......Page 277
Rank determines semi-stable conductor......Page 279
References......Page 286
Introduction......Page 287
Elliptic curves and modular L-functions......Page 289
Dirichlet L-functions......Page 291
L-functions of function fields over finite fields......Page 293
Problems......Page 296
References......Page 299
Introduction......Page 301
Bounds for the star discrepancy of (0,1)-sequences......Page 304
Prerequisites......Page 305
The proofs of the results......Page 311
References......Page 313
Introduction......Page 315
The class field tower......Page 317
Modular parametrizations......Page 319
Degeneracy maps......Page 320
Canonical factorizations of isogenies......Page 321
Geometric action......Page 323
Proof of Theorem 1......Page 324
References......Page 326
On an additive representation function......Page 327
References......Page 334
Introduction......Page 335
The generalized Riemann-Roch problem......Page 339
The rank-one case......Page 341
The quadric surface example......Page 345
Acknowledgements......Page 351
Further Reading......Page 352
Introduction......Page 353
The classical Lehmer Conjecture......Page 354
Lehmer’s Conjecture in abelian extensions......Page 355
Preliminaries......Page 356
Local height functions......Page 357
The unramified case......Page 360
A congruence in ramified abelian extensions......Page 362
Ramified points in E(Kab)......Page 363
Proof of the main theorem......Page 365
Generalization to abelian varieties......Page 369
Acknowledgements......Page 370
References......Page 371
Introduction......Page 373
Fourier analysis......Page 374
Eisenstein series......Page 375
Main theorem......Page 376
References......Page 377
Corrigendum to ’’Coefficients of half-integral weight modular forms’’ [J. Number Theory 99 (2003) 164-179]......Page 378
index.pdf......Page 380