The Lifted Root Number Conjecture for small sets of places and an - - PowerPoint PPT Presentation

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 CM-extensions

Contents

1

Motivation Analytic and arithmetic objects Galois module structure

2

The Conjecture The arithmetic side The analytic side Proven cases Small sets of places

3

CM-extensions The LRNC on minus parts Related conjectures

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions Analytic and arithmetic objects Galois module structure

Contents

1

Motivation Analytic and arithmetic objects Galois module structure

2

The Conjecture The arithmetic side The analytic side Proven cases Small sets of places

3

CM-extensions The LRNC on minus parts Related conjectures

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions Analytic and arithmetic objects Galois module structure

The Riemann zeta function

The Riemann zeta function ζ(s) =

∞

• n=1

1 ns =

• p

1 1 − p−s , ℜ(s) > 1 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

• f rational numbers.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions Analytic and arithmetic objects Galois module structure

The Dedekind zeta function I

Let L be any number field

• L = ring of integers of L

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions Analytic and arithmetic objects Galois module structure

The Dedekind zeta function I

Let L be any number field

• L = ring of integers of L

The Dedekind zeta function ζL(s) =

• P⊳oL prime

1 1 − N(P)−s , ℜ(s) > 1 is attached to L, where N(P) = |oL/P|.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions Analytic and arithmetic objects Galois module structure

The Dedekind zeta function II

ζL(s) has an analytic continuation to C \ {1} and a simple pole at s = 1 with residue 2r1(2π)r2hLRL wL

• |dL|

. Here, r1 resp. r2 denotes the number of real resp. pairs of complex conjugate embeddings of L, wL is the number of roots of unity in L, dL the discriminant, RL the regulator and hL the class number of L.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions Analytic and arithmetic objects Galois module structure

Contents

1

Motivation Analytic and arithmetic objects Galois module structure

2

The Conjecture The arithmetic side The analytic side Proven cases Small sets of places

3

CM-extensions The LRNC on minus parts Related conjectures

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions Analytic and arithmetic objects Galois module structure

The structure of units

Aim: Describe the structure of the units o×

L .

Theorem (Dirichlet’s unit theorem) Write µL for the roots of unity of L. Then we have an isomorphism of Z-modules

• ×

L ≃ µL × Zr1+r2−1.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions Analytic and arithmetic objects Galois module structure

The structure of units

Aim: Describe the structure of the units o×

L .

Theorem (Dirichlet’s unit theorem) Write µL for the roots of unity of L. Then we have an isomorphism of Z-modules

• ×

L ≃ µL × Zr1+r2−1.

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 ZG-module!

Many further objects attached to L are equipped with a natural action of G.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

Contents

1

Motivation Analytic and arithmetic objects Galois module structure

2

The Conjecture The arithmetic side The analytic side Proven cases Small sets of places

3

CM-extensions The LRNC on minus parts Related conjectures

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases 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 oL S is called a finite set of places of L.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases 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 oL S is called a finite set of places of L. We want to admit denominators and define

• S =

a b|a, b ∈ oL, P ∤ (b) ∀P ∈ S

• .

The S-units ES are the invertible elements of oS: ES := o×

S .

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The Tate-sequence

If S is “sufficiently” large, there exists an exact sequence (“Tate-sequence”) ES ֌ A → B ։ ∆S with a uniquely determined extension class in Ext2

G(∆S, ES);

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The Tate-sequence

If S is “sufficiently” large, there exists an exact sequence (“Tate-sequence”) ES ֌ A → B ։ ∆S with a uniquely determined extension class in Ext2

G(∆S, ES);

∆S is the kernel of the augmentation-map ZS ։ Z,

• P∈S

xPP →

• P∈S

xP. ∆S has easy G-module structure.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The Tate-sequence

If S is “sufficiently” large, there exists an exact sequence (“Tate-sequence”) ES ֌ A → B ։ ∆S with a uniquely determined extension class in Ext2

G(∆S, ES);

∆S is the kernel of the augmentation-map ZS ։ Z,

• P∈S

xPP →

• P∈S

xP. ∆S has easy G-module structure. B is ZG-projective and can be chosen ZG-free. A is cohomologically trivial, i.e. there is a short exact sequence P1 ֌ P0 ։ A with ZG-projective P0 and P1.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The arithmetic object

There exist equivariant embeddings φ : ∆S ֌ ES with finite cokernel. One can transpose φ to an embedding ˜ φ : B ֌ A with finite cokernel; this cokernel is cohomologically trivial.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The arithmetic object

There exist equivariant embeddings φ : ∆S ֌ ES 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) Ωφ := (cok ˜ φ) − correction term ∈ K0T(ZG). K0T(ZG) is the free abelian group generated by isomorphism classes of finite cohomologically trivial ZG-modules modulo short exact sequences.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

Contents

1

Motivation Analytic and arithmetic objects Galois module structure

2

The Conjecture The arithmetic side The analytic side Proven cases Small sets of places

3

CM-extensions The LRNC on minus parts Related conjectures

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases 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

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases 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. Vχ = the FG-module attached to χ. ˇ Vχ = HomF(Vχ, F) = contragredient of Vχ with character ˇ χ. R(G) = free abelian group generated by the irreducible characters of G.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The Stark-Tate regulator II

The map Rφ : R(G) − → C× χ → det(λS ◦ φ|HomG( ˇ Vχ, C ⊗ ∆S)) is called the Stark-Tate regulator, where λS : ES − → R ⊗ ∆S ε → −

• P∈S

log |ε|PP is the Dirichlet map.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

L-functions

For a prime ideal P of L let φP ∈ G be the Frobenius automorphism and IP ≤ G the inertia subgroup at P. The series LS(L/K, χ, s) =

• p∈S(K)

det(1 − φPN(p)−s|V IP

χ )−1,

converges for s ∈ C, ℜ(s) > 1.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

L-functions

For a prime ideal P of L let φP ∈ G be the Frobenius automorphism and IP ≤ G the inertia subgroup at P. The series LS(L/K, χ, s) =

• p∈S(K)

det(1 − φPN(p)−s|V IP

χ )−1,

converges for s ∈ C, ℜ(s) > 1. LS(L/K, χ, s) has a meromorphic continuation to C. We denote the leading coefficient of the Taylor expansion at s = 0 by cS(χ).

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

L-functions

For a prime ideal P of L let φP ∈ G be the Frobenius automorphism and IP ≤ G the inertia subgroup at P. The series LS(L/K, χ, s) =

• p∈S(K)

det(1 − φPN(p)−s|V IP

χ )−1,

converges for s ∈ C, ℜ(s) > 1. LS(L/K, χ, s) has a meromorphic continuation to C. We denote the leading coefficient of the Taylor expansion at s = 0 by cS(χ). Example For L = K, S = S∞ and χ = 1 we have LS∞(K/K, 1, s) = ζK(s). Especially for K = Q, we have LS∞(Q/Q, 1, s) = ζ(s), cS∞(1) = ζ(0) = −1

2.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

Stark’s Conjecture and the LRNC

Conjecture (Stark) The map R(G) − → C× χ → Rφ(ˇ χ) cS(ˇ χ) W (ˇ χ) lies in HomΓ(R(G), F ×). Here, W (ˇ χ) = ±1 is defined via Artin root numbers.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

Stark’s Conjecture and the LRNC

Conjecture (Stark) The map R(G) − → C× χ → Rφ(ˇ χ) cS(ˇ χ) W (ˇ χ) lies in HomΓ(R(G), F ×). Here, W (ˇ χ) = ±1 is defined via Artin root numbers. If Stark’s Conjecture holds, the above homomorphism defines an element Θφ ∈ K0T(ZG).

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

Stark’s Conjecture and the LRNC

Conjecture (Stark) The map R(G) − → C× χ → Rφ(ˇ χ) cS(ˇ χ) W (ˇ χ) lies in HomΓ(R(G), F ×). Here, W (ˇ χ) = ±1 is defined via Artin root numbers. If Stark’s Conjecture holds, the above homomorphism defines an element Θφ ∈ K0T(ZG). Conjecture (LRNC, K.W. Gruenberg, J. Ritter, A. Weiss 1999) Θφ = Ωφ.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

Strong Stark Conjecture

Choose a maximal order M: ZG ⊂ M ⊂ QG. There is a natural map δM : K0T(ZG) → K0T(M) induced by ⊗ZG M.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

Strong Stark Conjecture

Choose a maximal order M: ZG ⊂ M ⊂ QG. There is a natural map δM : K0T(ZG) → K0T(M) induced by ⊗ZG M. Conjecture (Strong Stark) Assume that Stark’s Conjecture holds. Then δM(Θφ) = δM(Ωφ).

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

Strong Stark Conjecture

Choose a maximal order M: ZG ⊂ M ⊂ QG. There is a natural map δM : K0T(ZG) → K0T(M) induced by ⊗ZG M. Conjecture (Strong Stark) Assume that Stark’s Conjecture holds. Then δM(Θφ) = δM(Ωφ). This conjecture is independent of the choice of M.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

Contents

1

Motivation Analytic and arithmetic objects Galois module structure

2

The Conjecture The arithmetic side The analytic side Proven cases Small sets of places

3

CM-extensions The LRNC on minus parts Related conjectures

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

Proven cases

K = Q and L/Q abelian (D. Burns,

• C. Greither 2003; for L totally real also by
• J. Ritter, A. Weiss 2002/03)

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

Proven cases

K = Q and L/Q abelian (D. Burns,

• C. Greither 2003; for L totally real also by
• J. Ritter, A. Weiss 2002/03)

L/K with E ⊂ K ⊂ L such that E = Q( √ d), where d < 0, hE = 1 and L/E is abelian, and such that [L : K] is odd and divisible only by primes which split completely in E/Q. (W. Bley 2006)

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

Contents

1

Motivation Analytic and arithmetic objects Galois module structure

2

The Conjecture The arithmetic side The analytic side Proven cases Small sets of places

3

CM-extensions The LRNC on minus parts Related conjectures

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

A Tate sequence for small sets of places

Actually, we are interested in the units: o×

L = ES∞.

Let the only constraint on S be: S∞ ⊂ S. Then there is a Tate sequence (J. Ritter, A. Weiss 1996) ES ֌ A → B ։ ∇.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

A Tate sequence for small sets of places

Actually, we are interested in the units: o×

L = ES∞.

Let the only constraint on S be: S∞ ⊂ S. Then there is a Tate sequence (J. Ritter, A. Weiss 1996) ES ֌ A → B ։ ∇. The torsion submodule of ∇ is the S class group clS(L): clS(L) ֌ ∇ ։ ∇ where ∇ is Z-free.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

A Tate sequence for small sets of places

Actually, we are interested in the units: o×

L = ES∞.

Let the only constraint on S be: S∞ ⊂ S. Then there is a Tate sequence (J. Ritter, A. Weiss 1996) ES ֌ A → B ։ ∇. The torsion submodule of ∇ is the S class group clS(L): clS(L) ֌ ∇ ։ ∇ where ∇ is Z-free. ∇ is the kernel in an exact sequence ∇ ֌ ZS ⊕

• P∈S∗

ram\(S∩S∗ ram)

ind G

GPW 0 P ։ Z

with explicit ZGP-modules W 0

P.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The conjecture for small sets of places I

Main problem: In general there are no embeddings φ : ∇ ֌ ES.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The conjecture for small sets of places I

Main problem: In general there are no embeddings φ : ∇ ֌ ES. Solution: There exist QG-isomorphisms φ : Q ⊗ ∇

≃

− → Q ⊗ (ES ⊕ C) with a ZG-free module C of suitable rank.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The conjecture for small sets of places I

Main problem: In general there are no embeddings φ : ∇ ֌ ES. Solution: There exist QG-isomorphisms φ : Q ⊗ ∇

≃

− → Q ⊗ (ES ⊕ C) with a ZG-free module C of suitable rank. From this one can construct QG-isomorphisms ˜ φ : Q ⊗ B

≃

− → Q ⊗ (A ⊕ C). Define Ωφ := (B, ˜ φ, A⊕C)− correction term ∈ K0(ZG, Q) ≃ K0T(ZG).

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The conjecture for small sets of places II

Determine how Ωφ varies if S is enlarged by (orbits of) prime ideals. Most interesting (and difficult to handle) are the primes which ramify in L/K. Unramified primes behave as before.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The conjecture for small sets of places II

Determine how Ωφ varies if S is enlarged by (orbits of) prime ideals. Most interesting (and difficult to handle) are the primes which ramify in L/K. Unramified primes behave as before. This leads to the definition of a modified Stark-Tate regulator Rmod

φ

.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The conjecture for small sets of places II

Conjecture (LRNC for small S) The element Ωφ ∈ K0(ZG, Q) is represented by the homomorphism R(G) → C×, χ → Rmod

φ

(ˇ χ) cS∪Sram(ˇ χ)W (ˇ χ).

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The arithmetic side The analytic side Proven cases Small sets of places

The conjecture for small sets of places II

Conjecture (LRNC for small S) The element Ωφ ∈ K0(ZG, Q) is represented by the homomorphism R(G) → C×, χ → Rmod

φ

(ˇ χ) cS∪Sram(ˇ χ)W (ˇ χ). The LRNC for small sets of places is equivalent to the LRNC for large sets of places. The LRNC naturally decomposes in local conjectures at each prime p by means of the isomorphism K0(ZG, Q) ≃

• p

K0(ZpG, Qp)

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Contents

1

Motivation Analytic and arithmetic objects Galois module structure

2

The Conjecture The arithmetic side The analytic side Proven cases Small sets of places

3

CM-extensions The LRNC on minus parts Related conjectures

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

CM-extensions

Let L/K be a Galois CM-extension with Galois group G, i.e. L is totally complex K is totally real Complex conjugation defines an automorphism j ∈ G on L, central in G: jg = gj ∀g ∈ G

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

CM-extensions

Let L/K be a Galois CM-extension with Galois group G, i.e. L is totally complex K is totally real Complex conjugation defines an automorphism j ∈ G on L, central in G: jg = gj ∀g ∈ G Example The extensions Q(ζn)/Q are abelian CM-extensions, where ζn denotes a n-th root of unity. Each abelian extension of Q lies in such an extension.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Plus and minus parts

For a G-Modul M set M± = {m ∈ M|jm = ±m}. If M is a ZpG-module and p = 2, there is a natural decomposition M = M+ ⊕ M−.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Plus and minus parts

For a G-Modul M set M± = {m ∈ M|jm = ±m}. If M is a ZpG-module and p = 2, there is a natural decomposition M = M+ ⊕ M−. Accordingly, the LRNC decomposes into a plus and a minus part.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Plus and minus parts

For a G-Modul M set M± = {m ∈ M|jm = ±m}. If M is a ZpG-module and p = 2, there is a natural decomposition M = M+ ⊕ M−. Accordingly, the LRNC decomposes into a plus and a minus part. Stark’s conjecture is known to be true on minus parts.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Ray class groups I

Let T be a finite set of prime ideals of L. Then we denote by clT

L := {fractional idelas of L, coprime to all P ∈ T}

{(a)|a ∈ L, a ≡ 1 mod P ∀P ∈ T} the ray class group of L to the ray MT :=

P∈T P.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Ray class groups II

Theorem Let L/K be a Galois CM-extension with Galois group G and p = 2 a prime such that L/K is at most tamely ramified above p. Then there exist finite sets T of primes of L such that (clT

L )− ⊗ Zp is

cohomologically trivial.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Ray class groups II

Theorem Let L/K be a Galois CM-extension with Galois group G and p = 2 a prime such that L/K is at most tamely ramified above p. Then there exist finite sets T of primes of L such that (clT

L )− ⊗ Zp is

cohomologically trivial. Indeed, a slightly weaker condition on the primes above p suffices.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Ray class groups II

Theorem Let L/K be a Galois CM-extension with Galois group G and p = 2 a prime such that L/K is at most tamely ramified above p. Then there exist finite sets T of primes of L such that (clT

L )− ⊗ Zp is

cohomologically trivial. Indeed, a slightly weaker condition on the primes above p suffices. (clT

L )− ⊗ Zp defines a class in K0(ZpG−, Qp).

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Connection with LRNC

Theorem Let L/K be a Galois CM-extension with Galois group G and p = 2 a prime, such that L/K is at most tamely ramified above p. Then there exist finite sets T of primes of L such that the homomorphism χ → cS∞(ˇ χ)

• p∈T(K)

det(1 − φ−1

P N(p)|V IP χ )

represents the class of (clT

L )− ⊗ Zp in K0(ZpG−, Qp) if and only if

the LRNC at p holds on minus parts.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Connection with LRNC

Theorem Let L/K be a Galois CM-extension with Galois group G and p = 2 a prime, such that L/K is at most tamely ramified above p. Then there exist finite sets T of primes of L such that the homomorphism χ → cS∞(ˇ χ)

• p∈T(K)

det(1 − φ−1

P N(p)|V IP χ )

represents the class of (clT

L )− ⊗ Zp in K0(ZpG−, Qp) if and only if

the LRNC at p holds on minus parts. For the proof, we use the LRNC for a (small!) set S of places which contains only totally decomposed primes.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Abelian CM-extensions

Theorem If in addition G is abelian, the LRNC at p holds on minus parts for almost all p.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Abelian CM-extensions

Theorem If in addition G is abelian, the LRNC at p holds on minus parts for almost all p. The main ingredient of the proof is the equivariant Iwasawa main conjecture. This conjecture is verified for abelian G (A. Wiles 1990 for p ∤ |G|, general case by

• J. Ritter/A. Weiss 2002).

The “descent” uses methods of A. Wiles in an extended version of C. Greither.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Strong Stark Conjecture

Proposition (Ritter/Weiss) Let L/K be a Galois extension of number fields and p a prime. Assume that Stark’s conjecture holds. Then the Strong Stark Conjecture at p holds if and only if the Strong Stark Conjecture holds for each cyclic intermediate extension of degree prime to p.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Strong Stark Conjecture

Proposition (Ritter/Weiss) Let L/K be a Galois extension of number fields and p a prime. Assume that Stark’s conjecture holds. Then the Strong Stark Conjecture at p holds if and only if the Strong Stark Conjecture holds for each cyclic intermediate extension of degree prime to p. The last Theorem + CM-version of the above proposition implies Corollary Let L/K be any Galois CM-extension. Then the Strong Stark Conjecture at p holds on minus parts for almost all primes p.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Contents

1

Motivation Analytic and arithmetic objects Galois module structure

2

The Conjecture The arithmetic side The analytic side Proven cases Small sets of places

3

CM-extensions The LRNC on minus parts Related conjectures

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Strong Brumer-Stark I

Let L/K be an abelian CM-extension and S and T finite sets of primes of L such that Sram ∪ S∞ ⊂ S T = ∅ S ∩ T = ∅ {ζ ∈ µL|ζ ≡ 1 mod P ∀P ∈ T} = 1

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Strong Brumer-Stark I

Let L/K be an abelian CM-extension and S and T finite sets of primes of L such that Sram ∪ S∞ ⊂ S T = ∅ S ∩ T = ∅ {ζ ∈ µL|ζ ≡ 1 mod P ∀P ∈ T} = 1 Define a Stickelberger element θT

S :=

• p∈T(K)

(1 − φ−1

P N(p))(

• χ irr.

LS(L/K, ˇ χ, 0)eχ) ∈ ZG where eχ =

1 |G|

• g∈G χ(g−1)g.

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Strong Brumer-Stark II

Let R = ZG− = ZG/(1 + j). Chose a resolution Ra

h

− → Rb ։ (clT

L )−

and define the Fitting ideal FittR((clT

L )−) = b × b − minors of hR

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Strong Brumer-Stark II

Let R = ZG− = ZG/(1 + j). Chose a resolution Ra

h

− → Rb ։ (clT

L )−

and define the Fitting ideal FittR((clT

L )−) = b × b − minors of hR

Remark: FittR((clT

L )−) ⊂ AnnR((clT L )−).

Conjecture (Strong Brumer-Stark) θT

S ∈ FittR((clT L )−)

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Overview

L/K CM-extension, abelian LRNC Strong Brumer-Stark

• ⇓

ETNC Rubin-Stark = ⇒ Brumer-Stark ⇓ ⇓ Strong Stark = ⇒ Stark Brumer ETNC = equivariant Tamagawa Number Conjecture

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Overview

L/K CM-extension, abelian LRNC Strong Brumer-Stark

• ⇓

ETNC = ⇒ Rubin-Stark = ⇒ Brumer-Stark ⇓ ⇓ Strong Stark = ⇒ Stark Brumer ETNC = equivariant Tamagawa Number Conjecture

• D. Burns 2007: ETNC =

⇒ Rubin-Stark

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Overview

L/K CM-extension, abelian LRNC Strong Brumer-Stark

• ⇓

ETNC = ⇒ Rubin-Stark = ⇒ Brumer-Stark ⇓ ⇓ Strong Stark = ⇒ Stark Brumer ETNC = equivariant Tamagawa Number Conjecture

• D. Burns 2007: ETNC =

⇒ Rubin-Stark

• C. Greither, M. Kurihara (preprint): wildly ramified

counter-examples of Strong Brumer-Stark

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

Overview

L/K CM-extension, abelian and tame LRNC = ⇒ Strong Brumer-Stark

• ⇓

ETNC = ⇒ Rubin-Stark = ⇒ Brumer-Stark ⇓ ⇓ Strong Stark = ⇒ Stark Brumer ETNC = equivariant Tamagawa Number Conjecture

• D. Burns 2007: ETNC =

⇒ Rubin-Stark

• C. Greither, M. Kurihara (preprint): wildly ramified

counter-examples of Strong Brumer-Stark

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

What next?

Generalization to non-abelian G assuming the validity of the equivariant Iwasawa main conjecture

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

What next?

Generalization to non-abelian G assuming the validity of the equivariant Iwasawa main conjecture On plus parts:

Stark’s Conjecture LRNC

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

What next?

Generalization to non-abelian G assuming the validity of the equivariant Iwasawa main conjecture On plus parts:

Stark’s Conjecture LRNC Does LRNC predict annihilators of the class group?

Andreas Nickel LRNC

Motivation The Conjecture CM-extensions The LRNC on minus parts Related conjectures

What next?

Generalization to non-abelian G assuming the validity of the equivariant Iwasawa main conjecture On plus parts:

Stark’s Conjecture LRNC Does LRNC predict annihilators of the class group?

On minus parts:

How to prove LRNC on minus parts for all primes p (p = 2)? Find a new descent method

Andreas Nickel LRNC