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Motivation The Conjecture CM-extensions The Lifted Root Number Conjecture for small sets of places and an application to CM-extensions Andreas Nickel Universit e de Bordeaux 1 Andreas Nickel LRNC Motivation The Conjecture


  1. Motivation The Conjecture CM-extensions The Lifted Root Number Conjecture for small sets of places and an application to CM-extensions Andreas Nickel Universit´ e de Bordeaux 1 Andreas Nickel LRNC

  2. Motivation The Conjecture CM-extensions Contents Motivation 1 Analytic and arithmetic objects Galois module structure The Conjecture 2 The arithmetic side The analytic side Proven cases Small sets of places CM-extensions 3 The LRNC on minus parts Related conjectures Andreas Nickel LRNC

  3. Motivation Analytic and arithmetic objects The Conjecture Galois module structure CM-extensions Contents Motivation 1 Analytic and arithmetic objects Galois module structure The Conjecture 2 The arithmetic side The analytic side Proven cases Small sets of places CM-extensions 3 The LRNC on minus parts Related conjectures Andreas Nickel LRNC

  4. Motivation Analytic and arithmetic objects The Conjecture Galois module structure CM-extensions The Riemann zeta function The Riemann zeta function ∞ 1 1 � � 1 − p − s , ℜ ( s ) > 1 ζ ( s ) = n s = n =1 p has an analytical continuation to C \ { 1 } and a simple pole at s = 1 with residue 1. The Riemann zeta function is attached to the number field Q of rational numbers. Andreas Nickel LRNC

  5. Motivation Analytic and arithmetic objects The Conjecture Galois module structure CM-extensions The Dedekind zeta function I Let L be any number field o L = ring of integers of L Andreas Nickel LRNC

  6. Motivation Analytic and arithmetic objects The Conjecture Galois module structure CM-extensions The Dedekind zeta function I Let L be any number field o L = ring of integers of L The Dedekind zeta function 1 � ζ L ( s ) = 1 − N ( P ) − s , ℜ ( s ) > 1 P ⊳ o L prime is attached to L , where N ( P ) = | o L / P | . Andreas Nickel LRNC

  7. Motivation Analytic and arithmetic objects The Conjecture Galois module structure CM-extensions The Dedekind zeta function II ζ L ( s ) has an analytic continuation to C \ { 1 } and a simple pole at s = 1 with residue 2 r 1 (2 π ) r 2 h L R L . � | d L | w L Here, r 1 resp. r 2 denotes the number of real resp. pairs of complex conjugate embeddings of L , w L is the number of roots of unity in L , d L the discriminant, R L the regulator and h L the class number of L . Andreas Nickel LRNC

  8. Motivation Analytic and arithmetic objects The Conjecture Galois module structure CM-extensions Contents Motivation 1 Analytic and arithmetic objects Galois module structure The Conjecture 2 The arithmetic side The analytic side Proven cases Small sets of places CM-extensions 3 The LRNC on minus parts Related conjectures Andreas Nickel LRNC

  9. Motivation Analytic and arithmetic objects The Conjecture Galois module structure CM-extensions The structure of units Aim: Describe the structure of the units o × L . Theorem (Dirichlet’s unit theorem) Write µ L for the roots of unity of L. Then we have an isomorphism of Z -modules o × L ≃ µ L × Z r 1 + r 2 − 1 . Andreas Nickel LRNC

  10. Motivation Analytic and arithmetic objects The Conjecture Galois module structure CM-extensions The structure of units Aim: Describe the structure of the units o × L . Theorem (Dirichlet’s unit theorem) Write µ L for the roots of unity of L. Then we have an isomorphism of Z -modules o × L ≃ µ L × Z r 1 + r 2 − 1 . Whenever L / K is a Galois extension of number fields with Galois group G , each σ ∈ G maps units to units; hence G acts on o × L . Higher aim: Describe o × L as a Z G -module! Many further objects attached to L are equipped with a natural action of G . Andreas Nickel LRNC

  11. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places Contents Motivation 1 Analytic and arithmetic objects Galois module structure The Conjecture 2 The arithmetic side The analytic side Proven cases Small sets of places CM-extensions 3 The LRNC on minus parts Related conjectures Andreas Nickel LRNC

  12. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places S -units S ∞ = set of all infinite primes, i.e. of all real embeddings and all pairs of complex conjugate embeddings of L S = S ∞ ∪ finite G -invariant set of prime ideals of o L S is called a finite set of places of L . Andreas Nickel LRNC

  13. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places S -units S ∞ = set of all infinite primes, i.e. of all real embeddings and all pairs of complex conjugate embeddings of L S = S ∞ ∪ finite G -invariant set of prime ideals of o L S is called a finite set of places of L . We want to admit denominators and define � a � o S = b | a , b ∈ o L , P ∤ ( b ) ∀ P �∈ S . The S -units E S are the invertible elements of o S : E S := o × S . Andreas Nickel LRNC

  14. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places The Tate-sequence If S is “sufficiently” large, there exists an exact sequence (“Tate-sequence”) E S ֌ A → B ։ ∆ S with a uniquely determined extension class in Ext 2 G (∆ S , E S ); Andreas Nickel LRNC

  15. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places The Tate-sequence If S is “sufficiently” large, there exists an exact sequence (“Tate-sequence”) E S ֌ A → B ։ ∆ S with a uniquely determined extension class in Ext 2 G (∆ S , E S ); ∆ S is the kernel of the augmentation-map � � Z S ։ Z , x P P �→ x P . P ∈ S P ∈ S ∆ S has easy G -module structure. Andreas Nickel LRNC

  16. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places The Tate-sequence If S is “sufficiently” large, there exists an exact sequence (“Tate-sequence”) E S ֌ A → B ։ ∆ S with a uniquely determined extension class in Ext 2 G (∆ S , E S ); ∆ S is the kernel of the augmentation-map � � Z S ։ Z , x P P �→ x P . P ∈ S P ∈ S ∆ S has easy G -module structure. B is Z G -projective and can be chosen Z G -free. A is cohomologically trivial, i.e. there is a short exact sequence P 1 ֌ P 0 ։ A with Z G -projective P 0 and P 1 . Andreas Nickel LRNC

  17. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places The arithmetic object There exist equivariant embeddings φ : ∆ S ֌ E S with finite cokernel. One can transpose φ to an embedding ˜ φ : B ֌ A with finite cokernel; this cokernel is cohomologically trivial. Andreas Nickel LRNC

  18. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places The arithmetic object There exist equivariant embeddings φ : ∆ S ֌ E S with finite cokernel. One can transpose φ to an embedding ˜ φ : B ֌ A with finite cokernel; this cokernel is cohomologically trivial. One defines (K.W. Gruenberg, J. Ritter, A. Weiss 1999) Ω φ := ( cok ˜ φ ) − correction term ∈ K 0 T ( Z G ) . K 0 T ( Z G ) is the free abelian group generated by isomorphism classes of finite cohomologically trivial Z G -modules modulo short exact sequences. Andreas Nickel LRNC

  19. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places Contents Motivation 1 Analytic and arithmetic objects Galois module structure The Conjecture 2 The arithmetic side The analytic side Proven cases Small sets of places CM-extensions 3 The LRNC on minus parts Related conjectures Andreas Nickel LRNC

  20. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places The Stark-Tate regulator I Let L ⊂ F be a number field such that F / Q is a Galois extension with Galois group Γ and “sufficiently” large. Let χ be a character of G . Andreas Nickel LRNC

  21. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places The Stark-Tate regulator I Let L ⊂ F be a number field such that F / Q is a Galois extension with Galois group Γ and “sufficiently” large. Let χ be a character of G . V χ = the FG -module attached to χ . ˇ V χ = Hom F ( V χ , F ) = contragredient of V χ with character ˇ χ . R ( G ) = free abelian group generated by the irreducible characters of G . Andreas Nickel LRNC

  22. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places The Stark-Tate regulator II The map C × R φ : R ( G ) − → det( λ S ◦ φ | Hom G ( ˇ χ �→ V χ , C ⊗ ∆ S )) is called the Stark-Tate regulator, where λ S : E S − → R ⊗ ∆ S � ε �→ − log | ε | P P P ∈ S is the Dirichlet map. Andreas Nickel LRNC

  23. The arithmetic side Motivation The analytic side The Conjecture Proven cases CM-extensions Small sets of places L -functions For a prime ideal P of L let φ P ∈ G be the Frobenius automorphism and I P ≤ G the inertia subgroup at P . The series det(1 − φ P N ( p ) − s | V I P � χ ) − 1 , L S ( L / K , χ, s ) = p �∈ S ( K ) converges for s ∈ C , ℜ ( s ) > 1. Andreas Nickel LRNC

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