Algebraic number fields: (L-functions and Galois properties) proceedings of a symposium (2 to 12 September 1975 in the University of Durham

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Preface Contents Character theory and Artin L-functions - J. Martinet I. NON ABELIAN L-FUNCTIONS §1. Frobenius §2. Weber §3. Artin's first definition of L-functions §4. The general definition of non abelian L-functions §5. Some elementary remarks on the Artin conjecture REFERENCES (CHAPTER I) II. GALOIS ACTION ON ROOT NUMBERS §1. More on the Artin conductor §2. Local Gauss sums §3. The transfer §4. Local Galois Gauss sums §5. Galois action on Galois Gauss sums and root numbers (local theory) §6. Real valued characters §7. Global theory §8. Global induction formulae III. ORTHOGONAL AND SYMPLECTIC REPRESENTATIONS §1. Description of real valued characters §2. Induction theorems §3. Induction theorems for orthogonal characters §4. Some arithmetic properties of orthogonal characters §5 Induction theorems for symplectic characters. REFERENCES (II and III) 7 IV. EXERCISES (Prepared jointly with J.-P. Serre) Local constants - J. T. Tate (prepared in collaboration with C.J. Bushnell & M.J. Taylor) (=Collected Works of John Tate, Part II, pp.31-73) Introduction Notations §1. Root Numbers in the Abelian Case §2. Existence of Local Constants §3. Root Numbers of Orthogonal Representations REFERENCES [AT] Galois module structure - A. Fröhlich Part I. Theorems on Galois module structure §1. Background §2. The classgroup of a group ring §3. Resolvents, Galois Gauss sums and Module structure §4. Root numbers and Galois module structure §5. Examples §6. Conductors Part II. Resolvent theory §7. The basic connections §8. Change of field or of group §9. Kummer extensions §10. Outline proof of Theorem 9 §11. Relation to Artin Conductor §11a. Congruence and Signature properties Part III. Galois Gauss sums and Root numbers §12. Congruence properties of Galois Gauss sums §13. Properties of Galois Gauss sums and root numbers §14. The range of symplectic root numbers §15. Symplectic root numbers for wild extensions Appendix §16. Once more: Change of field or of group §17. Factorial behaviour of resolvent classes and module classes REFERENCES [F10] Modular forms of weight one and Galois representations - J.-P. Serre (prepared in collaboration with C.J. Bushnell) PART I §1 . Two-dimensional Galois representations §2 . Modular Forms §3. The Main Theorems §4. Proof of Theorem 2 §5. Applications PART II §6. Cohomology and Liftings §7. Dihedral Representations §8. Representations with Prime Conductor §9. Modular Forms of Weight One on Γ₀(p) REFERENCES [AT] [H,24] [Sp] p-adic L-functions and Iwasawa's theory - John Coates Introduction §1. The algebraic theory 1.1 Class field theory 1.2. The basic Iwasawa module 1.3. Kummer theory 1.4. p-adic residue formula §2. Stiekelberger ideals 2.1 The partial zeta functions 2.2. The norm congruence lemma 2.3. Integrality 2.4. The Stickelberger ideals. §3. Stickelberger's theorem 3.1. Gauss sums 3.2. Proof of Stickelberger's theorem §4. p-adic L-functions 4.1. Values of L-functions 4.2. Construction of the G(T, χ). 4.3. The p-adic L-functions §5. The main conjecture 5.1. The main conjecture 5.2. Non group-theoretic evidence for the main conjecture. 5.3. Group-theoretic evidence for the main conjecture 5.4. Proof of the main conjecture in special cases. 5.5. Consequences of the main conjecture Appendix 1. REFERENCES 8 21 Class Fields for Real Quadratic Fields and L-series at 1 - H.M. Stark §1. Introduction §2. The numerical evaluation of L-series §3. The form of our conjecture for K/k §4. Two numerical examples. REFERENCES On Conductors and Discriminants - A.M. Odlyzko §1. Introduction §2. Proofs §3. Description of tables REFERENCES TABLE 1. Lower Bounds for Discriminants TABLE 2. Totally Real Fields With Small Discriminants TABLE 3. Totally Complex Fields with Small Discriminants TABLE 4. Lower Bound for Conductors m=χ(1) 8 A Relation Between ζK(s) and ζK(s-1) for any Algebraic Number Field K - Audrey Terras §0. Introduction §1. Summary of Results §2. Fourier Expansions of Nonanalytic Eisenstein Series for GL₂ over K. REFERENCES Some Global Norm Density Results obtained from an Extended Čebotarěv Density Theorem - R. Odoni Introduction §1. The reduction of Problem 1 §2. An interesting special case and an unsolved problem REFERENCES A Survey of Class Groups of Integral Group Rings - Stephen V. Ullom Introduction §1. Definitions and formal properties of the locally free class group §2. Methods of computation §3. Numerical results §4. Cyclic p-groups REFERENCES E GRU M Ro W 2 H₈ - J. Martinet §1. ℤ[G]-modules §2. Quaternion fields §3. The invariant U_N §4. Some comutations of the invariant U_N §5. Proof of theorem 2. REFERENCES [M] Un contre-example a une conjecture de J. Martinet - Jean Cougnard §.I. Groupes non abeliens d'ordre pq §.II. Extensions et résolvantes de Lagrange §.III. Décomposition des resolvantes de Lagrange §.IV. Construction de l'extension BIBLIOGRAPHIE 8 A Stickelberger Condition on Galois module structure for Kummer extensions of Prime degree - Leon R. McCulloh §1. The Main Theorem §2. Description of the Class Group §3. Calculation of cl(0_L) §4. The Stickelberger Condition §5. Corollaries REFERENCES 4 17 Stickelberger without Gauss sums - A. Fröhlich §1. Introduction §2. Module theory §3. Kummer theory §4. The Stickelberger relations §5. A cohomological criterion REFERENCES Fields of class two and Galois cohomology - H. Koch REFERENCES On p-closed number fields and an analogue of Riemann’s existence theorem - Olaf Neumann §1. The main results §2. Proof of theorem 1 (sketch) §3. Some corollaries of theorem 1 and bibliographical remarks REFERENCES 6 Holomorphy of Quotients of Zeta-Functions - Robert W. van der Waall Introduction 1. R. Dedekind 2. E. Artin 3. R. Brauer 4. M. Ishida 5. K. Uchida and R. van der Waall 6. Epilogue REFERENCES 2 14 GLₙ - W. Casselman §1. Archimedean fields §2. Non-archimedean fields. §3. Global fields. §4. GL₂(ℚ) REFERENCES 13 25