Arithmetic of Diagonal Hypersurfaces over Finite Fields (London Mathematical Society Lecture Note Series, Series Number 209)

There is now a large body of theory concerning algebraic varieties over finite fields, and many conjectures in this area are of great interest to researchers in number theory and algebraic geometry. This book deals with the arithmetic of diagonal hypersurfaces over finite fields, with special focus on the Tate conjecture and the Lichtenbaum-Milne formula for the central value of the L-function. It combines theoretical and numerical work, and includes tables of Picard numbers. Although this book is aimed at experts, the authors have included some background material to help nonspecialists gain access to the results. Contents......Page 5 Acknowledgments......Page 7 Notation and conventions......Page 9 Introduction......Page 13 1 Twisted Jacobi sums......Page 23 2 Cohomology groups of V = Vn (c)......Page 37 3 Twisted Fermat motives......Page 41 4 The inductive structure and the Hodge and Newton polygons......Page 51 5 Twisting and the Picard number......Page 63 6 "Brauer numbers" of twisted Fermat motives......Page 73 7 Evaluating Q(V, T) at T = q_r......Page 89 8 The Lichtenbaum-Milne conjecture......Page 95 9.1 The case of composite m......Page 103 9.2 The plus norms......Page 107 9.3 Further questions......Page 108 A.1 A note on the computations......Page 111 A.2 Twisted Fermat motives and their invariants......Page 112 A.3 Picard numbers of V = V,,(c).........Page 116 A.4 "Brauer numbers" of twisted Fermat motives......Page 134 A.5 Global "Brauer numbers" of V = V,, (c)......Page 138 B How to compute the stable Picard number when m is prime......Page 171 Bibliography......Page 175 Index ......Page 179 There Is Now A Large Body Of Theory Concerning Algebraic Varieties Over Finite Fields, And Many Conjectures Exist In This Area That Are Of Great Interest To Researchers In Number Theory And Algebraic Geometry. This Book Is Concerned With The Arithmetic Of Diagonal Hypersurfaces Over Finite Fields, With Special Focus On The Tate Conjecture And The Lichtenbaum–milne Formula For The Central Value Of The L-function. It Combines Theoretical And Numerical Work, And Includes Tables Of Picard Numbers. Although This Book Is Aimed At Experts, The Authors Have Included Some Background Material To Help Non-specialists Gain Access To The Results. Fernando Q. Gouvêa & Noriko Yui. Includes Bibliographical References (p.163-166) And Index.