Valued hyperfields, truncated DVRs, and valued fields Junguk Lee - - PowerPoint PPT Presentation

Preliminaries Main theorems

Valued hyperfields, truncated DVRs, and valued fields

Junguk Lee

Institute of mathematics, Wroclaw University

Logic Colloquium 2018 Udine, Italy 23-28, July, 2018

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Preliminaries Main theorems

[1] P. Deligne. Les corps locaux de caract´ eristique p, limits de corps locaux de caract´ eristique 0. J.-N. Bernstein, P. Deligne, D. Kazhdan, M.-F. Vigneras, Representations des groupes reductifs sur un corps local, Travaux en cours, Hermann, Paris, 119-157, (1984). [2] M. Krasner. Approximation des corps valu´ es complets de caract´ eristique p = 0 par ceux caract´ eristique 0. 1957 Colloque d’alg` ebre sup´ erieure, tenu ´ a Bruxelles du 19 au 22 d´ ecembre. [3] J. Lee and W. Lee. On the structure of certain valued fields, preprint. [4] J. Tolliver. An equivalence between two approaches to limits of local fields, J. of Number Theory, (2016), 473-492.

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Preliminaries Main theorems

(K, ν, k, Γ, R, m) (K, ν) is a henselian valued field of mixed characteristic (0, p). k is perfect (K, ν) is finitely ramified. We have eν(p) is finite, where for each x ∈ R we define eν(x) := |{γ ∈ Γ| 0 < γ < ν(x)}|. Note eν(p) is the ramification index. If there is no confusion, we denote eν by e. π ∈ R is a uniformizer so that Γ is generated by ν(π). For each n > 0, Rn := R/mn, called the n-th residue ring. If (K, ν) is a discrete complete valued fields, we may assume that ν is normalized, that is, ν(p) = 1. And we still write ν for the unique extension of ν in K alg

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Preliminaries Main theorems

Main Goal : Let K1 and K2 be complete discrete valued fields of mixed characteristic with perfect residue fields. There is an positive integer N depending only

• n the ramification indices such that if the N-th valued hyper valued

fields of valued fields are isomorphic over p, i.e., HN(K1) ∼ =HN({p}) HN(K2) , then K1 and K2 are isomorphic, i.e., K1 ∼ = K2.

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Preliminaries Main theorems

For each n > 0, we define the n-th valued hyperfield of K Hn(K) := (K/(1 + mn); +H, −H, ·H; 0H, 1H; νH), where for α := [a], β := [b] ∈ K/(1 + mn), 0H := [0] and 1H := [1]; α +H β := {[x + y]| x ∈ a(1 + mn), y ∈ b(1 + mn)}; β := (−Hα) is the unique element such that 0H ∈ α + β; α ·H β := [ab]; and νH(α) := ν(a), If there is no confusion, we omit H.

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Preliminaries Main theorems

Definition Let K1 and K2 be valued fields. Fix n > 0.

1 A map f from Hn(K1) to Hn(K2) is called a homomorphism if for

α, β ∈ Hn(K1),

f (0) = 0 and f (1) = 1; f (αβ) = f (α)f (β); f (α + β) ⊂ f (α) + f (β); and ν1(α) < ν1(β) ⇔ ν2(f (α)) < ν2(f (β)).

We denote Hom(Hn(K1), Hn(K2)) for the set of homomorphisms from Hn(K1) to Hn(K2).

2 A homomorphism f from Hn(K1) to Hn(K2) is called over p if

f ([p]) = [p]. We denote HomZ(Hn(K1), Hn(K2)) for the set of homomorphisms from Hn(K1) to Hn(K2), which is over p.

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Preliminaries Main theorems

Let (K1, ν1, k1, Γ1) and (K2, ν2, k2, Γ2) be finitely ramified henselian valued fields of mixed characteristic with perfect residue fields. Let e1 and e2 be the ramification indices of K1 and K2. Set N =

• 1

if p |e1(, tamely ramified) e2(p)(1 + e2

1(p)) + 1

if p|e1(, wildly ramified) . Here, e2

1(p) = e1(e1(p)) is well-defined because K1 is of characteristic 0

and N is a subset of K1.

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Preliminaries Main theorems Main Theorem 1

Theorem Suppose (K1, ν1) and (K2, ν2) be complete discrete valued fields. Then there is a unique lifting map L : HomZ(HN(K1), HN(K2)) → Hom(K1, K2) such that for any homomorphism φ : HN(K1) → HN(K2), set g = L(φ), and we have There exists a representaive β of φ([π1]) which satisfies ν2(

• g(π1) − β
• > M(K1),

where M(K1) := max{ν1(π1 − σ(π1))| σ ∈ HomL1(K1, K alg

1 ), σ(π1) = π1}

and L1(⊂ K1) is the quotient field of the Witt ring W (k1). φred,1 ◦ H1 = H1 ◦ g, where φred,1 : H1(K1) → H1(K2) is the natural reduction map of φ and H1 is the natural projection map to the first valued hyperfield.

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Preliminaries Main theorems Main Theorem 1

K1 K2 H1(K1) H1(K2)

H1 g H1 φred,1

Example Let K1 = Q3( √ 3) and K2 = Q3(√−3). Since e1 = e2 = 2, N = 1. Set π1 = √ 3 and π2 = √−3. Note that the sets of Teichm¨ uller representatives of K1 and K2 are (S :=)S1 = S2 = {−1, 0, 1}. Consider a homomorphism f ∈ Hom(H1(K1), H1(K2)) by sending [ √ 3] → [√−3]. Note that for each a ∈ S, f ([a]) = [a]. We have that f : H1(K1) ∼ = H1(K2), [ √ 3] → [ √ −3] which is not over p. But K1 and K2 are never isomorphic.

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Preliminaries Main theorems Main Theorem 2

Theorem Let K1 and K2 be finitely ramified henselian valued fields of mixed characteristic (0, p) with perfect residue fields. The followings are equivalent:

1 K1 ≡ K2. 2 HN(K1) ≡HN({p}) HN(K1).

Furthermore, if we take N′ = e2(p)(1 + e2

1(p)) + 1,

3 (S. Basarab, L. and W.Lee)R1,N′ ≡ R2,N′ and Γ1 ≡ Γ2.

Example Let K1 = Q3( √ 3) and K2 = Q3(√−3). Then e1 = e2 = 2, and N = 1 and N′ = 3. We have that K1 ≡ K2 by the sentence ∃X(X 2 + 3 =)0, and so H1(K1) ≡H1({3}) H1(K2) and R1,3 ≡ R2,3. But R1,1 = R2,1 = F3.

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Preliminaries Main theorems

Grazie!

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