• 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, ifL/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∞), thenJ (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 withr(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, concerninge(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,Se+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 somen ≥ 1.
Theorem 2 (Snaith) For a negative integerr,
annZp[G]((K1−2r(L)/E)p)er+
⊆ annZp[G](K−2r(OL,S)p)
where E consists of the Stark elements in K1−2r(L)and er+ = 1(1 + (−1)rc).
If F• is a perfect chain complex of Zp[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 ∈annZp[G](H1(F•)),
det(F•)tg ⊆ annZp[G](H0(F•)),
where g is the minimal number of generators(over Zp[G]) for Hom(H1(F•), Qp/Zp) 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 K1−2r(L)p
and K−2r(OL,S)p in degrees 1 and 2 (Q.–L.)and zero elsewhere. An appropriate modifica-tion gives rise to a complex C with
(K1−2r(L)/E)p if i = 1
Further, Hom((K1−2r(L)/E)p, Qp/Zp) is gener-ated by one element over Zp[G], so we can takeg = 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 fieldsL0 ⊆ L1 ⊆ L2 ⊆ · · · where Ln = Q(ζmpn+1),with Iwasawa algebra Λ. Then indeed, there isa Λ-complex C∞ such that C ⊗Λ Zp[Gn] = Cnfor 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 andL/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 Zp[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.

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