ファイル置き場 Sendai Logic Homepage

p ^{進体の逆数学}

山崎 武

東北大大学院理学研究科

2010 ^年 7 ^月 2 ^{日 全体セミナー}

Outline of this talk:

1. Reverse Mathematics and the field of p-adic numbers. 2. Continuous functions on Z_p.

Themes of Reverse Mathematics:

Let τ be a mathematical theorem. Let S_τ be the weakest natural subsystem of second order arithmetic in which τ is provable.

I. Very often, the principal axiom of S_τ is logically equivalent to τ ( over RCA⁰).

II. Furthermore, only few subsystems of second order arithmetic arise in this way.

Such subsystems are

   ^RCA_{0, WKL0, ACA0, ATR0, Π}¹_1-CA0．

Among the previous 5 subsystems, the following will be dealt with:

(1) ^RCA0 = the basic axioms + Σ⁰₁ induction + ∆⁰₁-CA. (2) ^WKL0 = RCA⁰ + weak K¨onig’s lemma.

(3) ^ACA0 = RCA⁰ + Π⁰_∞-CA.

1 ^Q

and Z

Definition 1 The following definitions are made in RCA⁰. Suppose that p is prime.

(1) For n ∈ Z, ord_pn = max{m ∈ N : p^m|n}.

(2) ^For a/b ∈ Q, ord_pa/b = ord_pa − ord_pb where a, b ∈ Z and b ̸= 0.

(3) ^For r ∈ Q, the p-adic norm of r is given by

|r|_p =

{ p^−ordpr if r ̸= 0 0 ^if r = 0

Basic properties of ord_p and | · |_p are provable in RCA⁰, particularly, | · |_p is a non-Archimedean norm on Q:

|r₁ + r₂|_p ≤ max{|r₁|_p, |r₂|_p}, with equality if |r₁|_p ̸= |r₂|_p, etc.

Definition 2 ^A p-adic number is defined in RCA^{0 to be} a sequence of rational numbers 〈r_i : i ∈ N〉 such that

∀k, i(|r_i − r_i+k|_p ≤ p⁻ⁱ). Two p-adic numbers 〈r_i : i ∈ N〉 and _〈ri^′ : i ∈ N〉 are said to be equal if

∀k(|r_k − r_k^′ |_p ≤ p^−k+1). A p-adic number 〈r_i : i ∈ N〉 is a p-adic integer if r_i ∈ Z for all i ∈ N.

We shall use the symbols Q_p and Z_p informally to denote the set of all p-adic numbers and one of all p-adic integers, respectively.

Within RCA⁰, we can define +, ·, | · |_p on Q_p. Theorem 1 It is provable in RCA⁰ that

Q_p, 0, 1, +, ·, | · |_p, =

obeys all the axioms of a field with a non-Archimedean norm _{| · |p}.

Moreover, Z_p has one-to-one correspondence to {x ∈ Q_p : |x|_p ≤ 1}

Theorem 2 The following assertions are pairwise equivalent over RCA⁰.

(1) ^ACA0

(2) ^Q_p is complete w.r.t. _{| · |p}

Proof of (2) ⇒ (1). Letting f : N → N be a one-to-one function, consider a sequence c_n = ^∑

i≤n

p^f(i)_{. ✷}

Lemma 3 The following is provable in RCA⁰. The series

∞

∑

i=0

a_i ^{in Q}_p converges if and only if _〈ai : i ∈ N〉 is a null sequence.

Lemma 4 The following is provable in RCA⁰. Given x ∈ Q_p, we can effectively find a unique sequence

〈a_i : −m ≤ i < ∞〉 such that x =

∞

∑

i=−m

a_ip^{i and}

∀i(0 ≤ a_i < p).

Lemma 5 (Hensel’s Lemma) The following is provable in RCA0. Let _{f (x) ∈ Zp}[x]. Suppose that a₀ ∈ Z_p ^satisfies f (a₀) ≡ 0 mod p and f^′(a₀) ̸≡ 0 mod p. Then there exists a unique a ∈ Z_p such that f (a) = 0 and a ≡ a₀ mod p.

Lemma 6 (Hasse-Minkowski Theorem) The following is provable in RCA⁰. Let Q(x₁, . . . , x_n) be a quadratic form over Q. Then the equation Q(x₁, . . . , x_n) = 0 has a

non-trivial solution in Q if and only if it has a non-trivial solution in R and every Q_p.

2 Continuous Functions on Z

Lemma 7 The following is provable in RCA⁰. Let x ∈ Q_p and δ > 0 be a real number. Let

y ∈ D(x, δ) = {z ∈ Q_p : |z − x|_p < δ}. Then D(x, δ) = D(y, δ)

Definition 3 The following definition is made in RCA⁰. A continuous function _{f : Zp → Qp} is locally constant if for each _{x ∈ Zp} we can effectively find a basic open ball _Ux such that _{f ⌈Ux} is constant and _{x ∈ Ux}.

Lemma 8 The following is provable in RCA0. _{f : Zp → Qp} is locally constant if and only if there is a countable open covering 〈U_n : n ∈ N〉 such that f ⌈U_n is constant for each n ∈ N.

Theorem 9 The following assertions are pairwise equivalent over RCA⁰.

(1) ^WKL0

(2) ^If f : Z_p → Q_p is locally constant, then the image of _f is finite.

Proof of (2) ⇒ (1). The case that p = 2. Within RCA⁰. Suppose that weak K¨onig’s lemma does not hold and let T ⊆ 2^N be an infinite tree with no infinite path. Then {[ ^∑

i<lh(σ)

σ(i)2ⁱ] : σ is an end node of T } is a disjoint open covering of Z₂. Define a locally constant function

f : Z_p → Q_p by

f (x) = lh(σ) if x ∈ [ ^∑

i<lh(σ)

σ(i)2ⁱ].

The image of f is infinite. ✷

Theorem 10 The following assertions are pairwise equivalent over RCA⁰.

(1) ^WKL0

(2) Every continuous function _{f : Zp} → R is uniformly continuous.

(3) Every continuous function _{f : Zp} → R is bounded. (4) Every bounded, uniformly continuous function

f : Z_p → R has a supremum.

(5) Every bounded, uniformly continuous function f : Z_p → R which has a supremum, attains it.

Theorem 11 The following assertions are pairwise equivalent over RCA0.

(1) ^ACA0

(2) If f : Z_p → Q_p is partial continuous, then there is a unique continuous extension F of f such that the

domain of F is a closure of the domain of f .

(3) If f : Z_p → Q_p is partial uniformly continuous, then there is a unique continuous extension F of f such that the domain of F is a closure of the domain of f .

Lemma 12 The following is provable in RCA0. If a

continuous function f : N → Q_p has a modulus of uniform continuity, then there a unique continuous extension

F : Z_p → Q_p ^of f .

Theorem 13 The following assertions are pairwise equivalent over RCA⁰.

(1) ^WKL0

(2) Any continuous function _{f : Zp → Qp} has Mahler

expansion, that is, there is a sequence _〈ai : i ∈ N〉 such that

∞

∑

n=0

a_n^(x n

)

converges uniformly to _{f (x).}

References

[1] Svetlana Katok. p-adic Analysis Compared with Real. Student Mathematical Library Vol. 37, AMS, 2007. X + 152 pages. [2] Stephen G. Simpson. Subsystems of Second Order Arithmetic.

Perspectives in Mathematical Logic. Springer-Verlag, 1999. XIV + 445 pages.