Number Theory: Sailing on the Sea of Number Theory: Proceedings of the 4th China-Japan Seminar

Contents......Page 22
Preface......Page 6
Program......Page 14
1. The analytic continuation of multiple Dirichlet series......Page 24
2. An example of double Dirichlet series with a natural boundary......Page 27
3. Proof of Theorem 2.1......Page 31
4. Proof of Theorem 2.2......Page 33
5. An application to the Riesz mean......Page 37
6. The multiple case......Page 41
References......Page 44
1. Introduction......Page 47
2. Selmer groups......Page 49
3. Oddness of graphs......Page 51
4. New non-congruent numbers......Page 53
References......Page 60
1. Introduction......Page 62
2.1. Polynomial g(x)......Page 66
2.2. Upper bound for #E(p)......Page 68
2.3. Conjecture......Page 70
3.1. Structure of o as an η-module......Page 72
3.2. Relη and κ(η)......Page 82
3.3.1. Action of automorphisms......Page 86
3.3.2. Evaluation of κ(η)......Page 90
4.1. Case of η = id......Page 93
4.1.1. Case of real quadratic fields......Page 95
4.1.3. Case of non-cyclic abelian fields of degree 4......Page 96
4.1.4. Case of imaginary abelian fields of degree 4......Page 99
4.1.5. Case where F is the Galois closure of a real cubic field F0 with negative discriminant......Page 101
4.2. Case of complex conjugation......Page 104
4.2.1. Case of [F : Q] = 4......Page 105
4.2.2. Case of [F : Q] = 6......Page 106
4.3. Case where F is an imaginary abelian field with [F : Q] = 6 and the order of    Gal(F/Q) is 3......Page 107
5.1. Divisors of f(p)......Page 110
5.2. Structure of Galois group extended by roots of units......Page 112
References......Page 119
2. The starting point......Page 120
3. Elliptic modular forms......Page 122
4. Siegel modular forms of genus two......Page 126
References......Page 129
Shifted Convolution Sums of Fourier Coefficients of Cusp Forms Yuk-Kam Lau, Jianya Liu and Yangbo Ye......Page 131
1.1. The classical case: the Riemann zeta-function and Dirichlet L-functions......Page 132
1.2. L-functions of degree two......Page 133
1.3. Rankin-Selberg L-functions......Page 134
1.5. Notations......Page 135
2.1. Spectral theory of automorphic forms......Page 136
2.2. The Rankin-Selberg method and shifted convolution sums......Page 137
3. Variants of the circle method......Page 138
3.1. The δ-symbol method......Page 139
3.2. Jutila’s variant......Page 142
4. The spectral method......Page 145
5. The spectral method: meromorphic continuation to σ > 1/2......Page 148
6.1. Further meromorphic continuation to σ >  1/2......Page 153
6.2. Illustration for the proof of Theorem 1.1......Page 155
References......Page 156
1. Introduction......Page 159
2. Part I: Generic polynomials of degree 3......Page 160
2.1. Generic polynomials of degree 3......Page 161
2.2. The Splitting Field of R(t;X)......Page 163
2.3. A Criterion for Kt  Kt......Page 164
2.4. Notes on the Reducible Cases......Page 165
2.5. An Application: Parametrization of Unrami.ed Cyclic Cubic Extensions of Quadratic Fields......Page 166
3.1. Elliptic Curves of the Form w3 = u3 + au2 + bu + c......Page 168
3.2. Some facts on the Hessian curves......Page 171
3.3. Twists of Hessian Elliptic Curves (1)......Page 172
3.4. Twists of Hessian Elliptic Curves (2)......Page 174
3.5. Some cases of non-empty H (µ, t)[Q]......Page 175
References......Page 176
1.1. Modular Hyperbolas and Kloosterman Sums......Page 178
1.3. Acknowledgements......Page 180
2.1. Exponential and Character Sums......Page 181
2.2. Theory of Uniform Distribution......Page 182
2.3. Arithmetic Functions, Divisors, Prime Numbers......Page 184
3.1. Points on Ha,m in Intervals for All a......Page 186
3.2. Points on Ha,m in Intervals on Average Over a......Page 188
3.3. Points on Ha,m in Sets with Arithmetic Conditions......Page 192
4.1. Distances......Page 193
4.2. Convex Hull......Page 197
4.3. Visible Points......Page 200
5.2. Distribution of Angles in Some Point Sets......Page 203
5.4. Sato-Tate Conjecture in the “Vertical” Aspect......Page 204
5.6. Approximations by Sums of Two Rationals......Page 205
5.8. Computing Discrete Logarithms and Factoring......Page 206
6.1. Generalisations......Page 207
References......Page 208
1. Erd˝os-Heilbronn conjecture and the polynomial method......Page 213
2. Various sumsets with polynomial restrictions......Page 218
3. Snevily’s conjecture and additive theorems......Page 223
4. On a conjecture of Lev and related results......Page 227
5. Working with general abelian groups......Page 230
6. On value sets of polynomials......Page 233
References......Page 234
1. Preliminaries......Page 237
2. Assumptions......Page 240
3. Theorem......Page 242
4. A General Modular Relation associated to the Riemann Zeta-function......Page 246
5. A General Modular Relation associated to a product of the Riemann Zeta-function......Page 252
References......Page 257
1. Introduction......Page 260
2. -adic cae:   = p......Page 261
3. p-adic case......Page 262
References......Page 264
Index......Page 266