The P-Adic Solenoid

The p-adic solenoid

Jordan Bell [email protected] Department of Mathematics, University of Toronto November 19, 2014

1 Deﬁnition

We shall be speaking about locally compact abelian groups, and unless we say otherwise, by morphism we mean a continuous group homomorphism. n For p prime and n ∈ Z≥0, p Z is a closed subgroup of the locally compact abelian group R, and the quotient R/pnZ is a compact abelian group. For n m n ≥ m, let φn,m : R/p Z → R/p Z be the projection map, which is a morphism. n The compact abelian groups R/p Z and the morphisms φn,m are an inverse system, and the inverse limit is a compact abelian group denoted Tp, called the n p-adic solenoid, with morphisms φn : Tp → R/p Z. Because the maps φn,m : n m n 1 R/p Z → R/p Z are surjective, the maps φn : Tp → R/p Z are surjective. n Let πn : R → R/p Z be the projection map, which is a morphism. The projection maps πn are compatible with the inverse system φn,m, so there is a unique morphism π : R → Tp such that φn ◦ π = πn for all n ∈ Z≥0. If x, y ∈ R are distinct, then for suﬃciently large n we have πn(x) 6= πn(y). If π(x) = π(y) then πn(x) = φn(π(x)) = φn(π(y)) = πn(y), a contradiction. n Therefore π : R → Tp is injective. Furthermore, the maps πn : R → R/p Z 2 being surjective implies that the image π(R) is dense in Tp.

2 Pontryagin dual

If G is a locally compact abelian group, we denote by G∗ the collection of morphisms G → S1. We assign G∗ the coarsest topology such that for all g ∈ G, the map γ 7→ γ(x) is continuous G∗ → S1, and with this topology, G∗ is a locally compact abelian group, called the Pontryagin dual of G. If φ : G → H is a morpism of locally compact abelian groups, then φ∗ : H∗ → G∗ deﬁned by

φ∗(θ)(g) = θ(φ(g)), θ ∈ H∗, g ∈ G,

1Alain M. Robert, A Course in p-adic Analysis, Chapter 1, §4, p. 29. 2Luis Ribes and Pavel Zalesskii, Proﬁnite Groups, p. 7, Lemma 1.1.7.

1 ∗ ∗ is a morphism. Say φ is surjective, and φ (θ1) = φ (θ2) but that θ1 6= θ2. Then there is some h ∈ H such that θ1(h) 6= θ2(h). Since φ : G → H is surjective, there is some g ∈ G such that φ(g) = h. But then

∗ ∗ θ1(h) = θ2(φ(g)) = φ (θ1)(g) = φ (θ2)(g) = θ2(φ(g)) = θ2(h),

∗ contradicting θ1(h) 6= θ2(h). Therefore, if φ : G → H is surjective then φ : H∗ → G∗ is injective. Let 1 j = : j ∈ ⊂ , pn Z pn Z Q which with the discrete topology is a discrete abelian group.

1 n ∗ Theorem 1. For prime p and n ∈ Z≥0, the map Φn : pn Z → (R/p Z) deﬁned by 1 Φ (a)(x + pn ) = e2πiax, a ∈ , x + pn ∈ /pn , n Z pn Z Z R Z is an isomorphism of topological groups.

j n n n Proof. Write a = pk , j ∈ Z. If x + p Z = y + p Z, then x − y ∈ p Z, so x − y = pnk for some k ∈ Z. Then

j n j n 2πiax 2πi n (p k+y) 2πik+2πi n y 2πiay n Φn(a)(x+p Z) = e = e p = e p = e = Φn(a)(y+p Z), showing that Φn is well-deﬁned. Furthermore, one checks that indeed Φn(a) ∈ n ∗ 1 (R/p Z) for each a ∈ pn Z. It is apparent that Φn(a + b) = Φn(a) · Φn(b). Φ is continuous because 1 n n pn Z is discrete. If Φn(a) = Φn(b), this means that for all x + p Z ∈ R/p Z, e2πiax = e2πibx, equivalently, that (a − b)x ∈ Z for all x ∈ R, whence a = b. Thus Φn is injective. n ∗ 1 ∗ Let γ ∈ (R/p Z) . Deﬁne Γ : R → S by Γ = γ ◦ πn, so that Γ ∈ R . We take as given that because Γ ∈ R∗, there is some y ∈ R such that Γ(x) = e2πiyx for all x ∈ R. In particular, for x = pn, on the one hand

n n n Γ(p ) = γ(πn(p )) = γ(0 + p Z) = 1, and on the other hand n Γ(pn) = e2πiyp , n 1 so yp ∈ Z, i.e. y ∈ pn Z, and it follows that γ = Φn(y). Therefore Φn is surjective. The open mapping theorem for topological groups states that if G, H are locally compact groups, f : G → H is a surjective morphism, and G is σ-compact, then f is open. Z is discrete and countable, hence is σ-compact, so Φn is open. Therefore Φn is an isomorphism of topological groups.

2 n m Because the morphisms φn,m : R/p Z → R/p Z are surjective, the mor- ∗ m ∗ n ∗ phisms φn,m :(R/p Z) → (R/p Z) are injective. For m ≤ n, deﬁne ιm,n : 1 1 j j pn−mj 1 pm Z → pn Z by ι pm = pm = pn ∈ pn Z; this is an injective morphism. One checks that the following diagram commutes.

∗ φn,m (R/pmZ)∗ (R/pnZ)∗

Φm Φn

1 ιm,n 1 pm Z pn Z

1 The discrete groups pm Z and the morphisms ιm,n are a direct system. The localization of Z away from p is the abelian group j [1/p] = : j ∈ , m ∈ ⊂ . Z pm Z Z≥0 Q

We assign Z[1/p] the discrete topology. One proves that Z[1/p] with the maps 1 ιm : pm Z → Z[1/p] deﬁned by

j j ι = m pm pm

3 1 1 is the direct limit of this direct system. The direct system ιm,n : pm Z → pn Z n m is dual to the inverse system φn,m : R/p Z → R/p Z. It follows that the Pontryagin dual of the limit of either system is isomorphic as a topological group to the limit of the other system. That is,

∗ ∼ ∗ ∼ Tp = Z[1/p], (Z[1/p]) = Tp, as topological groups.

3 p-adic integers

n m For n ≥ m, let ψn,m : Z/p Z → Z/p Z be the projection map. With the discrete topology, Z/pnZ is a compact abelian group, as it is ﬁnite. Then n m ψn,m : Z/p Z → Z/p Z is an inverse system, and its inverse limit is a compact abelian group denoted Zp, called the p-adic integers, with morphisms ψn : n Zp → Z/p Z. Because the morphisms ψn,m are surjective, the morphisms ψn are surjective. n n Let λn : Z/p Z → R/p Z be the inclusion map. Then the morphisms n Λn = λn ◦ ψn : Zp → R/p Z are compatible with the inverse system φn,m : 3A direct limit of discrete abelian groups is the direct limit of abelian groups. On direct limits of abelian groups, cf. Luis Ribes and Pavel Zalesskii, Proﬁnite Groups, p. 15, Proposition 1.2.1.

3 n m R/p Z → R/p Z, so there is a unique morphism Λ : Zp → Tp such that φn ◦ Λ = Λn for all n ∈ Z≥0. Suppose that x, y ∈ Zp are distinct and that Λ(x) = Λ(y). It is a fact that there is some n such that ψn(x) 6= ψn(y). Because λn is injective, this implies that Λn(x) 6= Λn(y), and this contradicts that Λ(x) = Λ(y). Therefore Λ : Zp → Tp is injective. One proves that ker φ0 = Λ(Zp), so that

0 → Zp → Tp → R/Z → 0 is a short exact sequence of topological groups.4 It can be proved that for each m ∈ Z>0 such that gcd(m, p) = 1, the p-adic solenoid Tp has a unique cyclic subgroup of order m, and on the other hand that there is no element in Tp whose order is a power of p, namely, Tp has no p-torsion.5

4 Further reading

Garrett has written several notes on the p-adic solenoid.6 The p-adic solenoid occurs in several places in the books of Hofmann and Morris.7 For properties of the p-adic solenoid involving homological algebra, see the below references.8

4Alain M. Robert, A Course in p-adic Analysis, Chapter 1, Appendix, p. 55. 5Alain M. Robert, A Course in p-adic Analysis, Chapter 1, Appendix, pp. 55–56. 6Paul Garrett, Solenoids, http://www.math.umn.edu/~garrett/m/mfms/notes/02_ solenoids.pdf; Paul Garrett, Bigger diagrams for solenoids, more automorphisms, colimits, http://www.math.umn.edu/~garrett/m/mfms/notes/03_more_autos.pdf; Paul Garrett, The ur-solenoid and the adeles, http://www.math.umn.edu/~garrett/m/mfms/notes/04_ur_ solenoid.pdf 7Karl H. Hofmann and Sidney A. Morris, The Structure of Compact Groups, 2nd revised and augmented edition; Karl H. Hofmann and Sidney A. Morris, The Lie Theory of Connected Pro-Lie Groups. 8 For Ext(Z, Tp) see Jean Dieudonn´e, A History of Algebraic and Diﬀerential Topol- ogy, 1900 – 1960, p. 94; see also J. M. Cordier and T. Porter, Shape Theory: Cate- gorical Methods of Approximation, p. 83; and http://mathoverflow.net/questions/4478/ torsion-in-homology-or-fundamental-group-of-subsets-of-euclidean-3-space