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The fractional Galois ideal•
Relation to Stark units•
The higher K
-group situation of Snaith
A very early analytic-to-algebraic result is theanalytic class number formula: For a numberfield L
Question 1 If L/K is Galois with Galois groupG, can we find (non-trivial) elements α ∈
We assume from now on that L/K
A partial answer is provided by Stickelbergerelements:
In fact, a known case (the rank zero case) ofStark’s conjecture, of which more later, showsthat θL/K,S ∈
In the case K
= Q, Stickelberger’s Theoremplus work of Siegel (later generalized by Deligneand Ribet) gives:
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
Z Q → X ⊗
Then for a representation V
with charac-ter χ
, Rfχ ∈
is defined to be the determinant(over C) of
](V ∗, X ⊗
Z C) →
](V ∗, X ⊗
The fractional Galois ideal and Stark’s
Stark’s conjecture (abelian case) says exactlythat Af ∈
. We assume this is the casefrom now on. This is known, for example, ifL/
Q is abelian.
to be the Z[G
]-submodule of Q[G
) | α ∈
Definition 1 Define the fractional Galois idealto be J
) = Af If ⊆
] for any choiceof
]-module isomorphism f as above. (Thisis independent of the choice of f .)
(and for simplicity S
) = J
) as defined earlier.
) always contains the Stickelbergerelement. Further, if for n ≥
) = Z[G
What about characters whose L
-functions havehigher orders of vanishing? For characters withr
) = 1, the Stark units come into the pic-ture.
Theorem 1 For simplicity of statement, sup-pose r
) = 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, concerninge
an odd prime, let L/
Q be a cyclotomicextension of p
-power conductor, and let S
Proposition 1 With L/
Q and S as above,
In this case, e
(0) = e−
(1) = e
) = Z[G
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
We assume that the Quillen–Lichtenbaum con-jecture holds, so that the Chern classes fromhigher K
-groups of number fields to ´
an odd prime again, and let m
be aninteger prime to p
) for somen ≥
Theorem 2 (Snaith) For a negative integerr,
where E consists of the Stark elements in K
+ = 1(1 + (−
is a perfect chain complex of Zp
any finite abelian group at the moment)which is acyclic outside degrees 0 and 1 andhas finite homology otherwise, then given t ∈
is the minimal number of generators(over Zp
]) for Hom(H
) 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
in degrees 1 and 2 (Q.–L.)and zero elsewhere. An appropriate modifica-tion gives rise to a complex C
) is gener-ated by one element over Zp
], so we can takeg
= 1 in the previous slide.
) 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 fieldsL
0 ⊆ L
1 ⊆ L
2 ⊆ · · ·
+1),with Iwasawa algebra Λ. Then indeed, there isa Λ-complex C∞
such that C ⊗
] = Cn
for each n
1 = er
+ − er−θ∞
is a limit of Stickelberger elements.
A careful descent argument allows one to seethat er
1 = er
+ from this.
Take a pair (L/
) satisfying certain assump-tions, where E
is an elliptic curve over Q andL/
Q is a (finite) abelian extension.
A Stark conjecture exists for (L/
thanks to the Equivariant TamagawaNumber Conjecture of Burns and Flach.
] can be defined as be-fore.
This elliptic curve Stark conjecture is
integral in the sense of predicting spe-cial points on E
), which we will callStark points.
(Stark points) in a similar way tothe number field case.
From Stark points to the Selmer group
There is a complex C
acyclic outside degrees 1, 2 and 3,whose cohomology groups are E
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.
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