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§ 3 Additive structure

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Uchida’s theorem for one-dimensional function fields over finite fields

Koichiro Sawada

Research Institute for Mathematical Sciences, Kyoto University

2016/07/18

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.Theorem (Uchida) ..

...

(cf. [Uchida] Isomorphisms of Galois groups of algebraic function fields)

Let K: one-dim function field/finite field, Ω: solvably closed Galois ext. of K

(i.e., Gal. ext. of K which has no nontriv. abelian ext.) Then K can be reconstructed from Gal(Ω/K).

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§ 1 Local theory

In this section, let k: local field of char. p > 0.

Write Gk = Gal(ksep/k).

Let us reconstruct the multiplicative structure of k×, together with various objects arising from k.

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Notation

Ok k: ring of integers

Ok := Ok\ {0}: multiplicative monoid of nonzero integers

Uk(1): multiplicative group of principal units Ik Gk: inertia subgroup

Pk ⊂Ik: wild inertia subgroup κ: residue field of k

Frobκ Gal(κ/κ): Frobenius element

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(local class field theory)

1 // Ok× //

(k×) //

//

1

1 // Im(Ik →Gabk ) // Gabk // Gk/Ik // 1, where the right-hand arrow maps

1 7→Frobκ Gal(κ/κ) Gk/Ik k× = ⟨π⟩ × O×k = ⟨π⟩ ×(k×)tor×Uk(1)

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Local reconstruction

p: unique prime number l s.t.

l | ♯κ = ♯(k×)tor + 1 = ♯(Gabk )tor + 1 fk := [κ : Fp] = logp(♯(Gabk )tor+ 1) Ik = ∩

kksep, k/k: fin. unr.Gk

k/k: unr [k : k] = [κ : κ]

[Gk : Gk] = fk/fk

Pk: unique Sylow pro-p-subgroup of Ik

Frobκ Gal(κ/κ) Gk/Ik: unique element of Gk/Ik which acts by conjugation on Ik/Pk by pfk

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Im(O×k ,→k× ,→Gabk ) = Im(Ik →Gabk ) Im(k× ,→Gabk ): subgp of Gabk gen. by

Im(O×k ,→Gabk ) and (a lifting of) Frobκ (in Gabk ) Im(Ok ,→k× ,→Gabk ): submonoid of Gabk gen. by Im(O×k ,→Gabk ) and (a lifting of) Frobκ (in Gabk ) Uk(1): unique Sylow pro-p-subgroup of Ok×

Ok× ,→ Okκ ⊃κ× induce isoms of groups (O×k)tor → O k×/Uk(1) κ× (which determine an isom of fields (k ) (Ok×)tor ∪ {0} κ)

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§ 2 Reconstruction of the multiplicative structure (global case)

In the rest of this talk, let

K: one-dim function field/fin. field of char. p > 0, Ω: solvably closed Gal. ext. of K.

Write G := Gal(Ω/K).

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Local Global

V: set of all (nonarchimedean) places of Ω VK: set of all (nonarchimedean) places of K

V˜: set of all maximal closed subgps of G which are isom to the abs. Gal. group of a local field (G conj.V˜)

Then

V w 7→Dw V˜: bij, VK →V˜/G: bij

(by Neukirch’s work, i.e., by considering local and global Brauer groups “H2(Gal. group))

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J = lim−→S⊂VK: fin. subset(∏

vS Kv×)×(∏

v∈VK\SOK×v) : id`ele group

J →Gab is determined by Kv× ,→ Dvab Gab

K× = ker(J →Gab) (global class field theory)

Write Uv(1), Ov×, Ov ⊂K×: inv. image of UK(1)

v, OK×v, OKv ⊂Kv× by K× ,→ J Kv×

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Order

a K×, v ∈ VK, n Z Write

ordv(a) := 1 if Ov is gen. by Ov× and a as monoid ordv(a) := n if ∃b K× s.t. ordv(b) = 1 and

a·bn ∈ Ov×

(well-defined)

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Evaluation v ∈ VK, s ∈ Ov

κv = κ×v ∪ {0}: residue field of K at v Let us define s(v) κv as follows:

if s ∈ Ov×, then s(v): image of s by Ov×Ov×/Uv(1) = κ×v ,→ κv

if s /∈ Ov×, then s(v) := 0

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§ 3 Additive structure

We want to reconstruct the add. str. of K = K× ∪ {0}. First, we reconstruct the add. str. of residue fields.

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Write F: constant field of K Then

F× = ∩

v∈VK Ov×

p: unique prime number l s.t. l | ♯F×+ 1

1 K×: if p ̸= 2, then 1: unique element a K× s.t. a2 = 1 and a ̸= 1 if p= 2, then 1 = 1

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Write K˜ = K F F

1 Gal(Ω/K˜) G→ GF 1: exact Let H ⊂G: open subgp. Then

H ker(G ↠GF) = Gal(Ω/K)˜ [G: H] = [FH : F]

= log♯F(♯FH) (in this case, H corresponds to K F FH)

By taking limit for such H’s, we obtain the following data ( ˜K× ∪ {0}, VK˜, {ordv}v∈VK˜, {Ov κv}v∈VK˜) (Note: K = ( ˜K)G)

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Divisor Div := ⊕

v∈VK˜ Z·v Let D = ∑

v∈VK˜ nv ·v Div

(nv = 0 for all but finitely many v) Then

H0(D)

= {s K˜× | ordv(s) + nv 0 (∀v ∈ VK˜)} ∪ {0} l(D) = min{n∈ Z0 | ∃v1, . . . , vn ∈ VK˜ s.t.

H0(D −v1 − · · · −vn) ={0}}

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Additive structure of residue fields

Let us fix v ∈ VK˜ and reconstruct the add. str. of κv

∃D = ∑

w∈VK˜ nw·w Div, ∃w1, w2 ∈ VK˜ s.t.

v, w1, w2: distinct, nv = nw1 = nw2 = 0, l(D) = 2, l(D −v −w1) =l(D −v −w2) = l(D −w1 −w2) = 0

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Let ζ, λ κ×v s.t. ζ ̸= (1)·λ

! s H0(D) s.t. s(v) =ζ, s(w1) = 0, s(w2) ̸= 0

! t∈ H0(D) s.t. t(v) = λ, t(w1) ̸= 0, t(w2) = 0

! u H0(D) s.t. u(w1) = t(w1), u(w2) = s(w2)

ζ +λ = u(v) (since u coincides with s+t)

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Additive structure of K˜ Let x, y K˜

if x = 0, then x+y := y if y = 0, then x+y := x

if x = (1)·y, then x+ y := 0 if x ̸= 0, y ̸= 0, x ̸= (1)·y, then

x+y: unique element z K˜× which satisfies the following condition:

for infinitely many v ∈ VK˜ s.t. x, y, z ∈ O×v , it holds that x(v) +y(v) =z(v)

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.Remark ..

...

The Frobenius homomorphism Frob: K →K (which is not an isomorphism!) determines an isomorphism

GFrob: G← G.

K(G) K(G)

K(GFrob)

oo

K ̸∼

Frob //

OO

K

OO

̸↷

参照

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