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## Events.math.unipd.it

*• *Motivation

*• *Stickelberger

*• *The fractional Galois ideal

*• *Relation to Stark units

*• *Cyclotomic example

*• *The higher

*K*-group situation of Snaith

*• *Equivariant Birch–Swinnerton-Dyer
A very early analytic-to-algebraic result is theanalytic class number formula: For a numberfield

*L*,
annZ(

*µ*(

*L*))

*J *(

*L*)

*⊆ *annZ(Cl(

*L*))
Question 1

*If L/K is Galois with Galois groupG, can we find (non-trivial) elements α ∈ *Q[

*G*]

*such that*
annZ[

*G*](

*µ*(

*L*))

*α ⊆ *annZ[

*G*](Cl(

*L*))?
We assume from now on that

*L/K *is abelian.

A partial answer is provided by Stickelbergerelements:

*LL/K,S*(0

*, χ*)

*e*¯

*χ ∈ *C[

*G*]

*.*
In fact, a known case (the rank zero case) ofStark’s conjecture, of which more later, showsthat

*θL/K,S ∈ *Q[

*G*].

In the case

*K *= Q, Stickelberger’s Theoremplus work of Siegel (later generalized by Deligneand Ribet) gives:
annZ[

*G*](

*µ*(

*L*))

*θL/*Q

*,S ⊆ *annZ[

*G*](Cl(

*L*))

*.*
However,

*θL/K,S *is often zero.

The analytic class number formula provides apossible hint: Take leading coefficients of

*L*-functions and divide by regulators. But thenwe need a regulator for each character of theGalois group. This can be done (with a caveat).

The existence of an R[

*G*]-module isomorphism(namely Dirichlet’s regulator map)
implies the existence of a (non-canonical) Q[

*G*]-module isomorphism

*L,S ⊗*Z Q

*→ X ⊗*Z Q

*.*
Then for a representation

*V *of

*G *with charac-ter

*χ*,

*Rfχ ∈ *C

*× *is defined to be the determinant(over C) of
HomC[

*G*](

*V ∗, X ⊗*Z C)

*→ *HomC[

*G*](

*V ∗, X ⊗*Z C)
The fractional Galois ideal and Stark’s
Stark’s conjecture (abelian case) says exactlythat

*Af ∈ *Q[

*G*]

*×*. We assume this is the casefrom now on. This is known, for example, if

*L/*Q is abelian.

Define

*If *to be the Z[

*G*]-submodule of Q[

*G*]generated by

*{*detQ[

*G*](

*α*)

*| α ∈ *EndQ[

*G*](

*X⊗*ZQ)

*, α◦f*(

*O×*
Definition 1

*Define the fractional Galois idealto be J *(

*L/K, S*) =

*Af If ⊆ *Q[

*G*]

*for any choiceof *Q[

*G*]

*-module isomorphism f as above. (Thisis independent of the choice of f .)*
If

*L *=

*K *(and for simplicity

*S *=

*S∞*), then

*J *(

*L/L, S*) =

*J *(

*L*) as defined earlier.

*J *(

*L/K, S*) always contains the Stickelbergerelement. Further, if for

*n ≥ *0
then

*e*(0)

*J *(

*L/K, S*) = Z[

*G*]

*θL/K,S*.

What about characters whose

*L*-functions havehigher orders of vanishing? For characters with

*r*(

*χ*) = 1, the Stark units come into the pic-ture.

Theorem 1

*For simplicity of statement, sup-pose r*(

*χ*) = 1

*for all χ ∈ G. Then under minorassumptions on the set S,*
*where E is the group of Stark units.*
One can remove the assumption on the ordersof vanishing of the

*L*-functions and still ob-tain a statement in a similar vein, concerning

*e*(1)

*J *(

*L/K, S*).

Fix

*p *an odd prime, let

*L/*Q be a cyclotomicextension of

*p*-power conductor, and let

*S *=

*{∞, p}*.

Proposition 1

*With L/*Q

*and S as above,*
24annZ[

*G*](

*µ*(

*L*))

*J *(

*L/*Q

*, S*)

*⊆ *annZ[

*G*](Cl(

*L*))

*.*
*Proof: *In this case,

*e*(0) =

*e− *and

*e*(1) =

*e*+.

*e−J *(

*L/*Q

*, S*) = Z[

*G*]

*θL/*Q

*,S*
*e*+

*J *(

*L/*Q

*, S*) =
Work of Rubin on cyclotomic units (which areexamples of Stark units), relates these units tothe class-group, finishing the proof.

From Stark-type elements to

*K*-groups
We assume that the Quillen–Lichtenbaum con-jecture holds, so that the Chern classes fromhigher

*K*-groups of number fields to ´
Take

*p *an odd prime again, and let

*m *be aninteger prime to

*p *and

*L *= Q(

*ζmpn*) for some

*n ≥ *1.

Theorem 2 (Snaith)

*For a negative integerr,*
annZ

*p*[

*G*]((

*K*1

*−*2

*r*(

*L*)

*/E*)

*p*)

*er*+

*⊆ *annZ

*p*[

*G*](

*K−*2

*r*(

*OL,S*)

*p*)

*where E consists of the Stark elements in K*1

*−*2

*r*(

*L*)

*and er*+ = 1(1 + (

*−*1)

*rc*)

*.*
If

*F• *is a perfect chain complex of Z

*p*[

*G*]-modules(

*G *any finite abelian group at the moment)which is acyclic outside degrees 0 and 1 andhas finite homology otherwise, then given

*t ∈*annZ

*p*[

*G*](

*H*1(

*F•*)),
det(

*F•*)

*tg ⊆ *annZ

*p*[

*G*](

*H*0(

*F•*))

*,*
where

*g *is the minimal number of generators(over Z

*p*[

*G*]) for Hom(

*H*1(

*F•*)

*, *Q

*p/*Z

*p*) and det(

*F•*)is the determinant of Knudsen and Mumford.

There exists a (cochain) complex to which theabove can be applied and which contains all thenecessary arithmetic information. It starts outas an ´
etale complex with cohomology

*K*1

*−*2

*r*(

*L*)

*p*
and

*K−*2

*r*(

*OL,S*)

*p *in degrees 1 and 2 (Q.–L.)and zero elsewhere. An appropriate modifica-tion gives rise to a complex

*C *with
(

*K*1

*−*2

*r*(

*L*)

*/E*)

*p *if

*i *= 1
Further, Hom((

*K*1

*−*2

*r*(

*L*)

*/E*)

*p, *Q

*p/*Z

*p*) is gener-ated by one element over Z

*p*[

*G*], so we can take

*g *= 1 in the previous slide.

*• *det(

*C*) is not found directly – Iwasawa the-oretic techniques are employed to deduce itfrom the determinant of a complex over anIwasawa algebra. Consider the tower of fields

*L*0

*⊆ L*1

*⊆ L*2

*⊆ · · · *where

*Ln *= Q(

*ζmpn*+1),with Iwasawa algebra Λ. Then indeed, there isa Λ-complex

*C∞ *such that

*C ⊗*Λ Z

*p*[

*Gn*] =

*Cn*for each

*n*.

det(

*C∞*)

*−*1 =

*er*+

*− er−θ∞ *mod Λ

*×*
where

*θ∞ *is a limit of Stickelberger elements.

*• *A careful descent argument allows one to seethat

*er*+det(

*Cn*)

*−*1 =

*er*+ from this.

Take a pair (

*L/*Q

*, E*) satisfying certain assump-tions, where

*E *is an elliptic curve over Q and

*L/*Q is a (finite) abelian extension.

*• *A Stark conjecture exists for (

*L/*Q

*, E*)
thanks to the Equivariant TamagawaNumber Conjecture of Burns and Flach.

*J *(

*L/*Q

*, E*)

*⊆ *Q[

*G*] can be defined as be-fore.

*• *This elliptic curve Stark conjecture is
integral in the sense of predicting spe-cial points on

*E*(

*L*), which we will callStark points.

*E*(

*L*)

*/*(Stark points) in a similar way tothe number field case.

*From Stark points to the Selmer group*
*• *There is a complex

*C *of Z

*p*[

*G*]-modules,
acyclic outside degrees 1, 2 and 3,whose cohomology groups are

*E*(

*L*)

*p*,Sel(

*E/L*)

*p *and (

*E*(

*L*)tors)

*∨p *in degrees 1,2 and 3.

*• *This would be modified to introduce the

*• *A similar result to earlier deals with per-
fect complexes acyclic outside

*three *ad-jacent cohomology groups, relating theannihilator ideals of the outside two co-homology groups to that of the middleone.

Source: http://events.math.unipd.it/aag/slides/slides-buckingham.pdf

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