Calculation of general Gauss sums and quadratic Gauss sums in finite rings

Chapters in this book

1. I-IV I
2. Introduction V
3. Remerciements / Acknowledgements IX
4. Liste des participants / List of participants XI
5. Table des matières / Table of contents XVII
6. Exponential Diophantine equations 1
7. Quantitative aspects of the representations of integers by quadratic forms 7
8. Diagonal cubic equations, I 15
9. Irrational dilations of approximate multiplicative semigroups of integers 29
10. Some special features of the Jacobi group 41
11. Généralisation de suites de Pisot et de suites de Boyd 51
12. Salem numbers of degree four have periodic expansions 57
13. Applications of partition theory to groups: Covering the alternating group by products of conjugacy classes 65
14. Variation of local heights on an algebraic family of abelian varieties 72
15. Sur une fausse forme modulaire liée à des identités de Ramanujan et Andrews 97
16. Global orders in definite quaternion algebras as endomorphism rings for reduced CM elliptic curves 108
17. Distribution properties of continuous curves 117
18. On the torsion in quotients of the multiplicative groups in abelian extensions 128
19. Binary quadratic forms over rings of algebraic integers: a survey of recent results 133
20. Places totalement décomposées dans des Zp-extensions d'un corps de nombres 160
21. Grandes valeurs de fonctions liées aux diviseurs premiers consécutifs d'un entier 169
22. The maximal order and the average order of multiplicative function σ(e)(n) 201
23. Discrépance quadratique de suites infinies en dimension un 207
24. Integral solutions of linear systems 213
25. Fractional parts of sequences 220
26. On the distribution of divisor class groups of curves over a finite field 227
27. Diophantine approximation, Kronecker's limit formula and the Riemann Hypothesis 240
28. Le théorème de Riemann-Roch et les corps de nombres algébriques 251
29. Functional equations and periodic sequences 267
30. Zp-extensions of function fields 280
31. Pseudoprimes on elliptic curves 290
32. Least primes in arithmetic progressions 306
33. Cyclic Galois extensions and normal bases 322
34. Cubic and biquadratic pseudoprimes of Lucas type 330
35. On some mean value results for the Riemann zeta-function 348
36. Classical explicit reciprocity 359
37. The distribution of badly approximable numbers and continuants with bounded digits 371
38. Approximation de périodes de fonctions méromorphes 386
39. L-series of automorphic forms on GL(3,R) 398
40. Equations differentielles Fuchsiennes et approximations diophantiennes 409
41. Exponential sums and the Riemann zeta function 417
42. Some recent results on the Riemann zeta-function 424
43. On the representations of a group as a Galois group over an arbitrary field 441
44. On evaluation of certain limits in closed form 459
45. Potential theory on curves 475
46. Characterization of arithmetical functions, problems and results 544
47. On a sufficient condition for the p—class tower of a CM—field to be infinite 556
48. Calculation of general Gauss sums and quadratic Gauss sums in finite rings 561
49. Transfinite diameter and Favard's problems on diameters of algebraic integers 574
50. Poles of Dirichlet series and D-modules 579
51. Small prime solutions of linear equations 595
52. Unusually large gaps between consecutive primes 625
53. On the congruence for Gauss sums and its applications 633
54. On central extensions 642
55. Prime producing quadratic polynomials and real quadratic fields of class number one 654
56. Disjoint systems of rational Beatty sequences 664
57. Complete determination of all torsion groups of elliptic curves with integral absolute invariant over quadratic and pure cubic fields 671
58. On generalized zeta—functions 699
59. Bounds for solutions of simultaneous diagonal equations of odd degree 703
60. Long arithmetic progressions and powers of 2 735
61. Sur la cohomologie de certains modules galoisiens p-ramifiés 740
62. On Euler's φ — function in quadratic number fields 755
63. Ternary code construction of even unimodular lattices 772
64. On the distribution of values of an error term related to the Euler function 785
65. Formes lineaires de logarithmes elliptiques et mesures de transcendance 798
66. On a question of Tverberg 806
67. Titchmarsh series 811
68. Galois representations associated to generic formal groups 815
69. A local construction of the local root numbers 823
70. On Fermat d-pseudoprimes 841
71. Exponential sums of digit counting functions 861
72. Some computational results on a problem of Einsenstein 869
73. The Eisenstein measure and real quadratic fields 887
74. Carmichael's conjecture and some analogues 928
75. The algebraic independence of three numbers: Schneider's method 942
76. The number of solutions of certain equations of the form f(xm) = g(yn) over a finite field 968
77. On entire functions assuming integer values in a geometric sequence 981
78. Stickelberger's theorem for cyclotomic fields, in the spirit of Kummer and Thaine 990
79. A propos d'un lemme de Κ. Kato sur le groupe de Brauer en caractéristique ρ 994
80. 1003-1006 1003