2. METODOLOGÍA
2.1. Componentes del robot
2.1.1. Placa base
integers of the division fields of F in the “Kummer” case, that is, considering
K(F[πm+r]) as an extension ofK(F[πr]) for integersr ≥m ≥1. Use the notation
M = K(F[πr]) and N = K(F[πm+r]), so that N/M is a Galois extension with group isomorphic to F[πm]. Specifically, we fix a primitive πm+r-torsion point α and define ψ : Γ = Gal(N/M) → F[πm] by ψ(γ) = γ(α)−
F α. This map is an
isomorphism known as the Kummer isomorphism. In [CNT, Ch. X], the authors consider the ring of integersON as an additive Γ-module. They determine that the associated order of this module inside MΓ is
AN/M =OM + qm−2
X
i=0
where σi = π1m
P
γ∈Γψ(γ)
i(γ−1).
We would like to make a comparison between their associated order and the elements of AF(N) described in Theorems 10.1 and 10.2. An obstacle to this com-
parison is that while the Galois module OM can use ON as its ring of scalars, a Lubin-Tate formal group only admits scalar multiplication by OK. Therefore we will compute the intersection of AN/M with KΓ.
Proposition 10.3. The intersection of AN/M with KΓ is generated as an OKΓ- module by τi = πm1−i
P
γ∈Hiγ for i= 0, . . . , m−1, where Hi = Gal(N/K([π
i+r])).
Proof. It is clear that τi ∈ KΓ for all i. Next we show that τi ∈ AN/M. Let
Mi =NHi, so thatM =M0 ⊂M1 ⊂ · · · ⊂Mm =N. Then we have AN/M ∩KΓ =
AN/M0 ∩KΓ ⊇ AN/M1 ∩KΓ ⊇ · · · ⊇AN/Mm ∩KΓ = OK. By Cassou-Nogu`es and
Taylor’s result, τi is an element of AN/Mm−i (it is the σ0 generator).
To finish the proof, we show that if φ is any element of AN/M ∩KΓ then the coefficients of the elements of Hi−Hi+1 ⊂Γ agree up to an element ofOK.
Ifθ ∈ Gal(M/K) then it restricts to an automorphism of F[πm]. Let θ
∗ be the
automorphism of Γ such that θ ◦ψ = ψ ◦θ∗. Using the correspondence between
θ and θ∗, the Galois group of M/K can act on φ ∈ MΓ in two ways—write θ·φ
for θ acting on the coefficients of φ and φθ for the action on elements of Γ via
θ−1
∗ . One checks that these two actions agree on σi for all i. Now if we act on
φ ∈AN/M∩KΓ we are guaranteed to have invariance under the left action because
θ fixes K. If we let φ =a+Pqm−2
i, we have φθ−φ=φθ−θ·φ=a−θ(a)∈ O
M, that is, φθ−φ∈ OM∩KΓ =OK. This proves the above claim, for we note that, since Gal(M/K) acts transitively on the set of primitive πi-torsion points, the θ
∗ act transitively on Hi −Hi+1 for i= 0, . . . , m−1.
The π2-torsion field, the subject of Theorem 10.1, is an instance of a Kummer
tower withr=m= 1. WithN =K(F[π2]) andM =K(F[π]), the proposition says thatAN/M∩KΓ is generated overOKΓ byτ0 =π−1
P
γ∈Γγ. This means that, if the
same eigenspace decomposition from Section 7.1 is applied to the additive Galois module ON, this τ0 will lie in the associated order of every eigenspace. However, for the points of F in ON, we have seen in the previous two sections that while τ0
does indeed lie in the associated order of F(m)(r) for r = 2, . . . , q−2, the same is
not true for r = q. We have found by computation of examples that τ0 may fail to be in the associated order of F(m)(q−1), but as yet we do not have a proof of a
Bibliography
[Ber] Anne-Marie Berg´e,A propos de l’ordre associ´` e `a l’anneau des entiers d’une extension, d’apr`es H. Jacobinski, S´eminaire de Th´eorie des Nombres de Bor- deaux (1971–1972), Exp. no. 8.
[Bon] Mikhail V. Bondarko, Cohomology of formal group moduli and deeply ram- ified extensions, Mathematical Proceedings of the Cambridge Philosophical Society 135 (2003), no. 1, 19–24.
[Byo] Nigel P. Byott,Integral Galois module structure of some Lubin-Tate exten- sions, Journal of Number Theory 77 (1999), no. 2, 252–273.
[CNT] Philippe Cassou-Nogu`es and Martin J. Taylor, Elliptic functions and rings of integers, Progress in Mathematics, vol. 66, Birkh¨auser, 1987.
[CW] John Coates and Andrew Wiles,On the conjecture of Birch and Swinnerton- Dyer, Inventiones Mathematicae39 (1977), no. 3, 223–251.
2nd ed., Translations of Mathematical Monographs, vol. 121, American Mathematical Society, 2002.
[Gre] Cornelius Greither,On Chinburg’s second conjecture for abelian fields, Jour- nal f¨ur die reine und angewandte Mathematik 479 (1996), 1–37.
[Haz] Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, 1978.
[Hus] Dale Husem¨oller, Elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 111, Springer-Verlag, 2004.
[Lan] Serge Lang, Cyclotomic fields I and II, combined 2nd ed., Graduate Texts in Mathematics, vol. 121, Springer-Verlag, 1990.
[Leo] Heinrich-Wolfgang Leopoldt, Uber die Hauptordnung der ganzen Elemente¨ eines abelschen Zahlk¨orpers, Journal f¨ur die reine und angewandte Mathe- matik 201 (1959), 119–149.
[Mar] Jacques Martinet,Anneau des entiers d’une extension galoisienne consid´er´e comme module sur l’alg`ebre du groupe de Galois, Colloque de Th´eorie des Nombres de Bordeaux (1969), Bulletin de la Soci´et´e Math´ematique de France, M´emoire 25 (1971), 123–126.
[Noe] E. Noether,Normalbasis bei K¨orpern ohne h¨ohere Verzweigung, Journal f¨ur die reine und angewandte Mathematik 167 (1932), 147–152.
[Ser] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, 1979.
[Sha] Romyar T. Sharifi, Galois module structure of local unit groups, Algebra & Number Theory 7 (2013), no. 1, 157–191.
[Sil] Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, 2009.
[Tho] Lara Thomas, On the Galois module structure of extensions of local fields, Actes de la Conf´erence “Fonctions L et Arithm´etique”, Publications Math´ematiques de Besan¸con. Alg`ebre et Th´eorie des Nombres (2010), 157– 194.
[Ull] Stephen V. Ullom,Integral normal bases in Galois extensions of local fields, Nagoya Mathematical Journal 39 (1970), 141–146.