# Totally disconnected groups, the p-adic numbers and the adèles

**Lesson HA-LCA-17.** Self-checked by the writing AI.

For a suitable totally disconnected group, each compactly supported locally constant function belongs to a finite Fourier problem. Compact open subgroups supply both the finite quotient and its exact Haar constants. We develop this reduction, prove how duality treats unions and limits, and identify the rational solenoid, including its path components.

All groups are Hausdorff. Unless otherwise specified, \(G\) is an arbitrary LCA group and \(\Gamma=\widehat G\). Our Fourier convention is
\[
 \widehat f(\gamma)=\int_G f(x)\overline{\gamma(x)}\,dx.       \tag{1}
\]
Write \(\mathbb Z_{\mathrm{prof}}=\varprojlim_n\mathbb Z/n\mathbb Z\), with indices ordered by divisibility. This notation distinguishes the profinite completion from the Pontryagin dual \(\widehat{\mathbb Z}=\mathbb T\).

The free readings are Igusa's [TIFR lecture notes, *Lectures on Forms of Higher Degree*, Chapter II, §1.2](https://mathweb.tifr.res.in/Documents/Publications/Lectures/tifr59.pdf); Bell's [*The profinite completion of the integers, the p-adic integers, and Prüfer p-groups*](https://jordanbell.info/LaTeX/mathematics/profinite/profinite.pdf); Dikranjan's [*Introduction to Topological Groups*, §§3.8 and 7.4](https://users.dimi.uniud.it/~dikran.dikranjan/ITG.pdf); and Burgos–Verjovsky's [*Adelic Ahlfors-Bers theory*, §2.1, exact arXiv:1603.05676v1](https://arxiv.org/pdf/1603.05676v1). Every assertion used from these readings is proved here or in an exact earlier programme proof. In particular, all arithmetic inputs below have already been proved in HA-LCA-10–12.

Written and checked by GPT-6 Astra (OpenAI), Ultra, October 2026. The new exposition is released under CC0. Separately linked prerequisite readings and their source packages retain their licences.

## 1. Small and large compact open subgroups

An LCA group is totally disconnected when its only connected subsets are singletons. Such a group has a neighbourhood base of compact open subgroups by [HA-LCA-12, Proposition 1.2](the-structure-of-locally-compact-abelian-groups.md#ha-lca-12-proposition-1-2). The condition on its dual has a different meaning.

<a id="ha-lca-17-theorem-1-1"></a>
**Theorem 1.1 — Large compact open subgroups.** The following are equivalent:

1. \(\widehat G\) is totally disconnected.
2. Every compact subset of \(G\) lies in a compact open subgroup.

Consequently the class
\[
 \mathcal T=\{G:G\text{ and }\widehat G
                    \text{ are totally disconnected}\}    \tag{2}
\]
is closed under Pontryagin duality. A group in \(\mathcal T\) has both arbitrarily small and arbitrarily large compact open subgroups.

**Proof.** Suppose \(\Gamma\) is totally disconnected. Choose a compact open subgroup \(U\subset\Gamma\), and set \(K=U^\perp\subset G\). It is compact open by [HA-LCA-10, Corollary 2.2](subgroups-quotients-and-annihilators.md#ha-lca-10-corollary-2-2). Closed-subgroup duality [HA-LCA-10, Theorem 2.1](subgroups-quotients-and-annihilators.md#ha-lca-10-theorem-2-1) identifies \(G/K\) with \(\widehat U\). Since \(U\) is compact totally disconnected, \(\widehat U\) is a discrete torsion group by [HA-LCA-12, Theorem 5.1](the-structure-of-locally-compact-abelian-groups.md#ha-lca-12-theorem-5-1).

A compact set \(C\subset G\) has finite image in the discrete quotient \(G/K\). The subgroup generated by finitely many torsion elements of an abelian group is finite: if their orders are \(n_1,\ldots,n_r\), every combination can be represented with the \(j\)-th coefficient between zero and \(n_j-1\). The inverse image of this finite subgroup is a finite union of \(K\)-cosets, hence compact and open, and contains \(C\).

Conversely, a basic identity neighbourhood in \(\Gamma\) imposes finitely many uniform bounds on compact subsets of \(G\). Their finite union is contained in some compact open \(K\). The compact open annihilator \(K^\perp\) lies in that neighbourhood, since all its characters are exactly one on \(K\). Thus compact open subgroups form an identity-neighbourhood base in \(\Gamma\). Their cosets are clopen, and they separate any two distinct points because \(\Gamma\) is Hausdorff. A connected subset cannot meet both a clopen coset and its complement, so it has at most one point.

The assertion about \(\mathcal T\) follows from [Pontryagin duality, HA-LCA-09, Theorem 2.1](the-pontryagin-duality-theorem.md#ha-lca-09-theorem-2-1); applying duality twice recovers \(G\). Small compact open subgroups come from total disconnectedness of \(G\), and large ones from what we have just proved for its dual. \(\square\)

## 2. Fourier transforms of cosets

Use the dual Haar measure fixed by [HA-LCA-07, Theorem 2.1](the-fourier-inversion-theorem-and-the-dual-haar-measure.md#ha-lca-07-theorem-2-1).

<a id="ha-lca-17-proposition-2-1"></a>
**Proposition 2.1 — Coset formulas and volumes.** For a compact open subgroup \(K\subset G\) and \(a\in G\),
\[
 \widehat{1_{a+K}}(\gamma)
   =\mu(K)\overline{\gamma(a)}\,1_{K^\perp}(\gamma),\qquad
 \widehat\mu(K^\perp)=\mu(K)^{-1}.                          \tag{3}
\]
If \(K\subset K'\) are compact open, then both indices below are finite and
\[
 [K':K]=[K^\perp:K'^\perp].                               \tag{4}
\]

**Proof.** Translation gives the phase in (3). The integral of \(\overline\gamma\) on \(K\) equals \(\mu(K)\) if \(\gamma|_K=1\). Otherwise choose \(k\in K\) with \(\gamma(k)\ne1\). Translation by \(k\) preserves the integral and also multiplies it by \(\overline{\gamma(k)}\), forcing it to be zero. This proves the transform formula.

The indicator lies in \(L^1\cap L^2\). The earlier Plancherel theorem [HA-LCA-08, Theorem 1.1](the-plancherel-theorem.md#ha-lca-08-theorem-1-1) gives
\[
 \mu(K)=\|1_K\|_2^2
       =\mu(K)^2\widehat\mu(K^\perp),
\]
proving the reciprocal volume. These volumes are finite and positive since the subgroups are compact and open. The quotients in (4) are discrete compact spaces and hence finite. If their cardinalities are \(N,M\), respectively, Haar invariance gives
\[
 \mu(K')=N\mu(K),\qquad
 \widehat\mu(K^\perp)=M\widehat\mu(K'^\perp).
\]
Substitute the reciprocal volumes to obtain \(N=M\). Thus the normalization at different compact open stages agrees. \(\square\)

## 3. Every test function belongs to a finite Fourier problem

For \(G\in\mathcal T\), let \(\mathcal S(G)\) be the complex-valued locally constant functions of compact support. These are the Schwartz–Bruhat functions in this setting.

<a id="ha-lca-17-lemma-3-1"></a>
**Lemma 3.1 — Finite stages.** Every \(f\in\mathcal S(G)\) is invariant under translation by some compact open subgroup \(K\). It is a finite linear combination of indicators of \(K\)-cosets, and its support lies in a compact open subgroup \(K'\) containing \(K\).

**Proof.** The nonzero set of a locally constant function is both open and closed: the function is constant on a neighbourhood of every point, including each zero. Hence that set equals its support. At every point of its compact support choose a compact open coset on which \(f\) is constant. Finitely many, say \(x_j+K_j\), cover the support. Put \(K=\bigcap_jK_j\), which is compact open. The support is invariant under \(K\), and so is \(f\) on it. Its complement is also \(K\)-invariant, where \(f=0\). Each \(K_j/K\) is finite by compactness, so the finite covering cosets split into finitely many \(K\)-cosets. This proves the asserted finite sum. Theorem 1.1 puts the compact set \(\operatorname{supp}f\cup K\) in a compact open \(K'\). For \(f=0\), choose any compact open \(K\) and take \(K'=K\). \(\square\)

<a id="ha-lca-17-theorem-3-2"></a>
**Theorem 3.2 — Finite-stage inversion and Plancherel.** For compact open \(K\subset K'\), the Fourier transform maps the \(K\)-invariant functions supported in \(K'\) bijectively onto the \(K'^\perp\)-invariant functions supported in \(K^\perp\). It is the finite Fourier transform on \(K'/K\), with the constants given below. Consequently
\[
 \mathcal F:\mathcal S(G)\longrightarrow\mathcal S(\Gamma)
 \text{ is bijective},\qquad
 f(x)=\int_\Gamma\widehat f(\gamma)\gamma(x)\,d\gamma,\qquad
 \|\widehat f\|_2=\|f\|_2.                                 \tag{5}
\]
Moreover \(\mathcal S(G)\) is dense in both \(L^1(G)\) and \(L^2(G)\).

**Proof.** Let \(H=K'/K\), \(N=|H|\), and \(c=\mu(K)\). Closed-subgroup duality identifies
\(\widehat H=K^\perp/K'^\perp\). Specifically, restrict a character of \(G\) trivial on \(K\) to \(K'\); its kernel as a restriction map is \(K'^\perp\), and all characters of \(K'/K\) extend by HA-LCA-10, Theorem 2.1. A \(K\)-invariant function supported in \(K'\) is a function \(F:H\to\mathbb C\). Proposition 2.1 gives
\[
 \widehat f(\gamma)=
 \begin{cases}
  c\displaystyle\sum_{u\in H}F(u)\overline{\gamma(u)},
                                      &\gamma\in K^\perp,\\
  0,                                  &\gamma\notin K^\perp .
 \end{cases}                                                   \tag{6}
\]
For \(\gamma\in K^\perp\) the value depends only on its class in \(\widehat H\). Conversely every function on the finite \(\widehat H\) occurs, since finite Fourier inversion [HA-LCA-01, Theorem 2.2](fourier-analysis-on-finite-abelian-groups.md#ha-lca-01-theorem-2-2) inverts the matrix in (6).

For \(x\in K'\), the inverse integral in (5) equals
\[
 \frac1{Nc}\sum_{\chi\in\widehat H}
       c\sum_{u\in H}F(u)\overline{\chi(u)}\chi(x)
 =F(x+K),                                                     \tag{7}
\]
because each coset of \(K'^\perp\) has dual Haar mass \(1/\mu(K')=1/(Nc)\), and finite character orthogonality gives the last equality. For \(x\notin K'\), the character \(\gamma\mapsto\gamma(x)\) is nontrivial on \(K'^\perp\), by double annihilation. Its integral on that compact subgroup is zero by the translation argument in Proposition 2.1. Integrating separately on the finitely many cosets in (6) therefore gives zero, which equals \(f(x)\). This proves inversion at every point directly from the finite problem.

The finite Parseval identity [HA-LCA-01, Theorem 2.1](fourier-analysis-on-finite-abelian-groups.md#ha-lca-01-theorem-2-1) gives
\[
 \|\widehat f\|_2^2
 =\frac{c^2}{Nc}\sum_{\chi\in\widehat H}
       \left|\sum_{u\in H}F(u)\overline{\chi(u)}\right|^2
 =c\sum_{u\in H}|F(u)|^2
 =\|f\|_2^2.                                                 \tag{8}
\]
Lemma 3.1 puts every test function in a stage. Apply that lemma also on \(\Gamma\in\mathcal T\). A dual stage \(A\subset B\) has the form
\(A=K'^\perp,\ B=K^\perp\) by taking \(K=B^\perp,\ K'=A^\perp\) and using double annihilation. Finite-stage surjectivity then proves surjectivity on the whole test space.

For density, let \(h\in C_c(G)\). Theorem 1.1 puts its support in a compact open \(K'\). Such an \(h\) is uniformly continuous by [PRE-HAAR, Lemma 1.1](../prerequisites/src/haar-measure.md#ha-lca-pre-haar-lemma-1-1). Choose a compact open \(K\subset K'\) so small that
\(|h(x+k)-h(x)|<\varepsilon\) for every \(x\in G,k\in K\). Set
\[
 h_K(x)=\mu(K)^{-1}\int_K h(x+k)\,dk.
\]
Haar invariance makes this \(K\)-invariant; hence it is locally constant. It vanishes outside \(K'\), and its uniform distance from \(h\) is at most \(\varepsilon\). For \(r=1,2\),
\(\|h_K-h\|_r\le\varepsilon\mu(K')^{1/r}\).
The density of \(C_c(G)\) in \(L^1(G)\), [PRE-INTEGRAL, Theorem 3.1](../prerequisites/src/integration-and-l1.md#ha-lca-pre-integral-theorem-3-1), and in \(L^2(G)\), [PRE-HILBERT, Proposition 6.2](../prerequisites/src/hilbert-spaces-and-unitary-representations.md#ha-lca-pre-hilbert-proposition-6-2), proves the claim, for the full Haar spaces without a sigma-finiteness restriction. \(\square\)

## 4. Duality for unions and inverse limits

An inverse limit is the subgroup of a product consisting of compatible tuples, with its subspace topology.

<a id="ha-lca-17-proposition-4-1"></a>
**Proposition 4.1 — Open unions.** Suppose \(G=\bigcup_iG_i\), where the open subgroups \(G_i\) form an upward-directed family and have their subspace topologies. Restriction is a topological isomorphism
\[
 \widehat G\ \cong\ \varprojlim_i\widehat{G_i}.               \tag{9}
\]

**Proof.** A compatible family of characters defines one character on the union: two definitions agree in any subgroup containing their two domains. It is continuous on each open \(G_i\), hence continuous on \(G\). This constructs the inverse of restriction.

Every compact \(C\subset G\) lies in one \(G_i\): the open cover by the \(G_i\) has a finite subcover, and directedness puts its finitely many groups in a common group. The set \(C\) remains compact in that group's subspace topology. Therefore a uniform bound on \(C\) in \(\widehat G\) is exactly a bound in one inverse-limit coordinate. In the other direction a compact set in any \(G_i\) is compact in \(G\), so restriction is continuous in the compact-open topologies. Basic neighbourhoods in the product topology involve only finitely many coordinates; their inverse images are the corresponding finite intersections of these compact-open conditions. This proves continuity in both directions. \(\square\)

<a id="ha-lca-17-proposition-4-2"></a>
**Proposition 4.2 — Compact inverse limits.** Let \((G_i,p_{ij})\) be an inverse system of compact abelian groups over a directed set, with surjective transition maps \(p_{ij}:G_j\to G_i\) for \(i\le j\). Its inverse limit \(G\) is compact, and every projection \(q_i:G\to G_i\) is onto. Put \(H_i=\ker q_i\). Then
\[
 \widehat G=\bigcup_i H_i^\perp
            \ \cong\ \varinjlim_i\widehat{G_i}              \tag{10}
\]
as discrete groups, the direct maps being the injective character pullbacks.

**Proof.** The product of the \(G_i\) is compact by [PRE-BANACH, Lemma 4.1](../prerequisites/src/banach-spectrum.md#ha-lca-pre-banach-lemma-4-1). Each compatibility condition \(p_{ij}(x_j)=x_i\) is closed because \(G_i\) is Hausdorff. Thus their intersection \(G\) is compact.

Fix \(i\) and \(a\in G_i\), and add the closed constraint \(x_i=a\). Any finite list of compatibility constraints, together with this one, is satisfiable. Choose an index \(k\) dominating all their indices and \(i\); choose \(b\in G_k\) with \(p_{ik}(b)=a\), using surjectivity; set each coordinate occurring in the constraints to its image of \(b\). Fill other coordinates arbitrarily. The compatibility laws give all the required equalities. The finite intersection property in the compact product therefore supplies a compatible tuple with \(i\)-coordinate \(a\). This proves surjectivity, including for uncountable directed systems.

Every identity neighbourhood in \(G\) contains some \(H_k\): choose a contained basic neighbourhood involving finitely many coordinates and take \(k\) dominating them. A character \(\chi\) is close to one on some such neighbourhood, and hence on \(H_k\). The small-arc lemma [HA-LCA-02, Lemma 1.2](characters-and-the-dual-group.md#ha-lca-02-lemma-1-2) makes \(\chi\) exactly one on \(H_k\). It therefore factors algebraically through \(q_k\). The factor is continuous because \(q_k\), a compact-to-Hausdorff surjection, is a closed quotient map. This proves the union in (10). Pullback is injective by surjectivity of \(q_i\); common refinements identify the union with the algebraic direct limit, since compatible homomorphisms on its stages glue uniquely on the union. All the duals in (10) are discrete by [HA-LCA-02, Theorem 2.1](characters-and-the-dual-group.md#ha-lca-02-theorem-2-1). Hence this is also the asserted topological identification. \(\square\)

<a id="ha-lca-17-proposition-4-3"></a>
**Proposition 4.3 — Residue limits and torsion duals.** For every prime \(p\),
\[
 \widehat{\mathbb Z_p}\cong
 \mathbb Z(p^\infty):=\mathbb Z[1/p]/\mathbb Z
 \cong\mathbb Q_p/\mathbb Z_p,\qquad
 \widehat{\mathbb Z(p^\infty)}\cong\mathbb Z_p.              \tag{11}
\]
The torsion groups in (11) have the discrete topology. Also
\[
 \mathbb Z_{\mathrm{prof}}\cong\prod_p\mathbb Z_p,\qquad
 \widehat{\mathbb Z_{\mathrm{prof}}}\cong\mathbb Q/\mathbb Z. \tag{12}
\]

**Proof.** On \(\mathbb Z/p^n\mathbb Z\), characters are
\(x\mapsto e^{2\pi iax/p^n}\), by finite cyclic duality [HA-LCA-02, Corollary 3.2](characters-and-the-dual-group.md#ha-lca-02-corollary-3-2). Pullback along reduction from level \(n+1\) sends \(a\bmod p^n\) to \(pa\bmod p^{n+1}\). Sending \(a\bmod p^n\) to \(a/p^n+\mathbb Z\) identifies these injective maps with inclusion of the corresponding subgroups of \(\mathbb Z[1/p]/\mathbb Z\). Proposition 4.2 proves the first dual identification. The union of those finite subgroups has the discrete topology, so Proposition 4.1, or Pontryagin duality, proves the reverse identification.

The construction of \(\mathbb Q_p\), its integral subgroup and fractional parts is given in full in [HA-LCA-10, Example 6.4](subgroups-quotients-and-annihilators.md#ha-lca-10-example-6-4). Each coset modulo \(\mathbb Z_p\) has a representative in \(\mathbb Z[1/p]\), and \(\mathbb Z[1/p]\cap\mathbb Z_p=\mathbb Z\); this gives the other isomorphism in (11). Its quotient topology is discrete because \(\mathbb Z_p\) is open.

Both assertions of (12), including the finite Chinese remainder calculation, compatibility of all residues, continuity of the product maps, and the direct-sum decomposition of \(\mathbb Q/\mathbb Z\), were proved in [HA-LCA-12, Example 7.2](the-structure-of-locally-compact-abelian-groups.md#ha-lca-12-example-7-2). Alternatively, Proposition 4.2 applied to all \(\mathbb Z/n\mathbb Z\) identifies the dual directly with \(\bigcup_n n^{-1}\mathbb Z/\mathbb Z=\mathbb Q/\mathbb Z\). \(\square\)

## 5. The rational solenoid and its leaves

<a id="ha-lca-17-lemma-5-0"></a>
**Lemma 5.0 — The profinite transversal.** The group \(\mathbb Z_{\mathrm{prof}}\) is compact, totally disconnected, metrizable, and homeomorphic to the Cantor space \(\{0,1\}^{\mathbb N}\). The integers embed densely in it, and it has no isolated points. For every positive integer \(n\), reduction modulo \(n\) is onto with kernel \(n\mathbb Z_{\mathrm{prof}}\), an open subgroup also homeomorphic to Cantor space.

**Proof.** The compactness, total disconnectedness and countable-product metrizability follow from its finite residue construction, as in HA-LCA-12, Example 7.2 and Theorem 5.1. Integers inject because an integer divisible by every positive integer is zero. For density, a basic nonempty open cylinder specifies finitely many compatible residues. Take a common multiple \(N\) of their moduli and an integer representing the tuple's residue modulo \(N\). That integer belongs to the cylinder.

Here is an explicit Cantor model. Restriction to the cofinal moduli \(n!\) identifies \(\mathbb Z_{\mathrm{prof}}\) with their residue inverse limit. Conversely a compatible family modulo \(n!\) defines the residue modulo any integer by reducing at a factorial divisible by it, independently of the choice; both maps are continuous by finite-coordinate dependence. If \(r_n\) is the representative in \(\{0,\ldots,n!-1\}\), compatibility gives uniquely
\[
 r_{n+1}=r_n+a_n n!,\qquad a_n\in\{0,\ldots,n\}.             \tag{13}
\]
Thus \(\mathbb Z_{\mathrm{prof}}\) is homeomorphic to
\(\prod_{n\ge1}\{0,\ldots,n\}\).

To identify this product with binary Cantor space without a classification theorem, code an alphabet of size \(m\ge2\) by the prefix words
\[
 0,\ 10,\ 110,\ \ldots,\ 1^{m-2}0,\ 1^{m-1}.
\]
Every infinite binary string starts with exactly one of these words. Code the \(n\)-th digit in (13) using the alphabet size \(n+1\), and concatenate. Conversely parse successively using those sizes. Every word has positive finite length, so this defines inverse maps on infinite sequences. Finite output prefixes depend on finitely many input digits, and conversely any fixed number of parsed digits depends on a finite binary prefix. The maps are continuous and inverse. If one uses the usual ternary Cantor set, the further map
\((b_n)\mapsto\sum_{n\ge1}2b_n3^{-n}\) is continuous and injective: a first differing digit at \(n\) contributes \(2\cdot3^{-n}\), while the later tail is at most \(3^{-n}\). It is a homeomorphism onto that set by compactness.

Every basic product cylinder leaves some digit unrestricted with at least two choices. Hence there are no isolated points; binary diagonalization also proves uncountability. Under the product isomorphism (12), the kernel of reduction modulo \(n\) is
\(\prod_p p^{v_p(n)}\mathbb Z_p\). Multiplication by \(n\) has exactly this image and is injective, since the \(\mathbb Q_p\) are fields of characteristic zero; its local factors away from \(p\) are units. These local facts are part of HA-LCA-10, Example 6.4. Therefore this kernel is \(n\mathbb Z_{\mathrm{prof}}\). Reduction is onto already on integers, and its kernel is open. Multiplication by \(n\), a continuous bijection from compact to Hausdorff onto this kernel, is a homeomorphism, proving the final assertion. \(\square\)

<a id="ha-lca-17-theorem-5-1"></a>
**Theorem 5.1 — The rational solenoid.** There are topological group isomorphisms
\[
 \widehat{\mathbb Q_d}\ \cong\
 \varprojlim_{n\mid m}\mathbb R/n\mathbb Z
 \ \cong\
 \Sigma=(\mathbb R\times\mathbb Z_{\mathrm{prof}})
               /\{(k,k):k\in\mathbb Z\}.                  \tag{14}
\]
The transition maps in the inverse limit are reduction modulo the smaller modulus. The quotient map \(q:\mathbb R\times\mathbb Z_{\mathrm{prof}}\to\Sigma\) gives the coordinate formula
\[
 q(t,z)\longmapsto(t-z_n+n\mathbb Z)_n,                    \tag{15}
\]
where \(z_n\) is any integer representative of \(z\bmod n\).

The group \(\Sigma\) is compact, connected, and not locally connected. For \(0<\varepsilon<1/2\),
\[
 q\bigl((-\varepsilon,\varepsilon)
                    \times n\mathbb Z_{\mathrm{prof}}\bigr)
\]
is an open neighbourhood homeomorphic to an interval times Cantor space. Its path components are the cosets of the dense subgroup \(q(\mathbb R\times\{0\})\), and their group is the uncountable abstract quotient \(\mathbb Z_{\mathrm{prof}}/\mathbb Z\). The injective map \(\mathbb R\to\Sigma\) defining this subgroup is not a topological embedding; each path component carries a real-line leaf topology distinct from its subspace topology.

**Proof.** The discrete group \(\mathbb Q\) is the directed union of the open subgroups \(n^{-1}\mathbb Z\). A character of \(n^{-1}\mathbb Z\) is uniquely of the form
\[
 k/n\longmapsto e^{2\pi i kt/n},\qquad t\in\mathbb R/n\mathbb Z,
\]
by the dual of \(\mathbb Z\). This parametrization is a homeomorphism: it is a continuous bijection of a compact circle onto the Hausdorff dual. Restriction from \(m^{-1}\mathbb Z\) to \(n^{-1}\mathbb Z\), for \(n\mid m\), reduces \(t\bmod m\) to \(t\bmod n\). Proposition 4.1 proves the first isomorphism.

The diagonal integer subgroup in (14) is discrete, since
\((-1/2,1/2)\times\mathbb Z_{\mathrm{prof}}\) meets it only at zero. It is closed by [HA-LCA-09, Lemma 1.2](the-pontryagin-duality-theorem.md#ha-lca-09-lemma-1-2). The quotient is Hausdorff and its quotient map is open by [HA-LCA-09, Lemma 6.0](the-pontryagin-duality-theorem.md#ha-lca-09-lemma-6-0). The compact set \([0,1]\times\mathbb Z_{\mathrm{prof}}\) maps onto it: subtract a diagonal integer to bring the real coordinate into \([0,1]\). Hence \(\Sigma\) is compact.

Formula (15) is continuous, additive, independent of the representatives, compatible under reduction, and invariant under adding \((k,k)\). To compute its kernel before quotienting, the \(n=1\) coordinate forces \(t=k\in\mathbb Z\), and all other coordinates then force \(z=k\) in the profinite group. Thus its kernel is exactly the diagonal integers. It is onto: for a compatible family \((w_n+n\mathbb Z)\), choose \(t\in\mathbb R\) representing its coordinate modulo one. Each \(t-w_n\) is then an integer modulo \(n\), and those integer residues are compatible, defining \(z\). Formula (15) produces the family. The induced continuous bijection from compact \(\Sigma\) to the Hausdorff inverse limit is a homeomorphism.

The group \(\mathbb Q_d\) is torsion-free. Its compact dual is connected by [HA-LCA-12, Theorem 5.1](the-structure-of-locally-compact-abelian-groups.md#ha-lca-12-theorem-5-1), proving connectedness. For an interval \(I\) of length less than one, the map \(q\) is injective on \(I\times\mathbb Z_{\mathrm{prof}}\): two representatives differing by a diagonal integer have real difference of absolute value less than one, so that integer is zero. Its restriction to any open subset of this product is open, since \(q\) is open on the full group. This proves the asserted product charts, using Lemma 5.0.

A connected subset of such a chart projects to a singleton in its totally disconnected profinite coordinate. It cannot be an open neighbourhood, because that coordinate has no isolated points. If \(\Sigma\) were locally connected, a connected open neighbourhood could be chosen inside this chart, a contradiction. Thus it is not locally connected.

We prove the path assertion explicitly. Given a continuous path \(c:[0,1]\to\Sigma\) beginning at zero, cover its image by the above interval-product charts. Their inverse images form an open cover of \([0,1]\). The finite-net and Lebesgue-number argument [HA-LCA-16, Lemma 3.0](almost-periodic-functions-and-the-bohr-compactification.md#ha-lca-16-lemma-3-0) gives a finite subdivision so that the image of each subinterval lies in one chart. Invert the chart there to lift that segment to \(\mathbb R\times\mathbb Z_{\mathrm{prof}}\). Its profinite coordinate is constant, since an interval is connected and the profinite group is totally disconnected. The first lift starts at a diagonal integer pair, which can be subtracted to start at \((0,0)\). At each subsequent junction, two representatives of the same point differ by a diagonal integer; adjust the whole next lift by that integer to match the preceding endpoint. Its constant profinite coordinate then remains zero. The finitely many continuous segments agree at their endpoints, so they glue to a continuous lift in \(\mathbb R\times\{0\}\). Thus \(c(1)\in q(\mathbb R\times\{0\})\). Conversely \(s\mapsto q(st,0)\) joins zero to \(q(t,0)\). Translation proves the claim for every component.

The real map is injective since \((t,0)=(k,k)\) forces \(k=t=0\). It is dense: integers \(k\) can approximate any \(z\in\mathbb Z_{\mathrm{prof}}\) by Lemma 5.0, and
\(q(t-k,0)=q(t,k)\to q(t,z)\).
Two points \(q(t,z),q(t',z')\) differ by an element of the real image exactly when \(z-z'\in\mathbb Z\), as follows directly from the diagonal equivalence relation. This identifies the component group with \(\mathbb Z_{\mathrm{prof}}/\mathbb Z\). It is uncountable: \(\mathbb Z_{\mathrm{prof}}\) is uncountable by Lemma 5.0, whereas a countable union of its countable integer cosets would be countable. In particular the real image is proper. A topological embedding of \(\mathbb R\) would make that image locally compact in the subspace topology and therefore closed by HA-LCA-09, Lemma 1.2, contradicting proper density.

For the suspension description, use representatives \(q(t,-z)\) with \(0\le t\le1\). Their endpoints obey
\[
 q(1,-z)=q(0,-z-1),
\]
so \((1,z)\) is identified with \((0,z+1)\). These are exactly the endpoint identifications, with no identifications for \(0<t<1\), by the same integer-difference calculation. The compact quotient of \([0,1]\times\mathbb Z_{\mathrm{prof}}\) by this relation maps bijectively and continuously to Hausdorff \(\Sigma\), hence homeomorphically. This is the suspension of translation \(z\mapsto z+1\). \(\square\)

## 6. Local fields and the rational adèles

The earlier programme already supplies the arithmetic needed here. In particular, [HA-LCA-10, Example 6.4](subgroups-quotients-and-annihilators.md#ha-lca-10-example-6-4) constructs the field \(\mathbb Q_p\) from residue limits, its compact open integer ring, its valuations and fractional parts, and proves its topological self-duality and Haar normalization. Thus none of the following identifications is an unproved preview.

<a id="ha-lca-17-proposition-6-1"></a>
**Proposition 6.1 — The \(p\)-adic balls.** Put
\(\psi_p(x)=e^{2\pi i\{x\}_p}\), and identify \(y\in\mathbb Q_p\) with the character \(x\mapsto\psi_p(xy)\). Then \(\mathbb Q_p\in\mathcal T\), and for the self-dual Haar measure with \(\mu(\mathbb Z_p)=1\),
\[
 (p^k\mathbb Z_p)^\perp=p^{-k}\mathbb Z_p,\qquad
 \widehat{1_{a+p^k\mathbb Z_p}}(y)
 =p^{-k}\psi_p(-ay)1_{p^{-k}\mathbb Z_p}(y).                \tag{16}
\]
In particular \(\widehat{1_{\mathbb Z_p}}=1_{\mathbb Z_p}\).

**Proof.** The exact earlier result cited above gives the self-duality, the compact open base \(p^k\mathbb Z_p\), and \(\mathbb Z_p^\perp=\mathbb Z_p\). Clopen cosets from this base separate points, so \(\mathbb Q_p\) is totally disconnected; self-duality puts it in \(\mathcal T\). Multiplication in the pairing gives
\[
 y\in(p^k\mathbb Z_p)^\perp
 \ \Longleftrightarrow\ p^ky\in\mathbb Z_p^\perp
 \ \Longleftrightarrow\ y\in p^{-k}\mathbb Z_p.
\]
The quotient \(\mathbb Z_p/p\mathbb Z_p\) has \(p\) elements. Its translated cosets have equal measure, so iterating gives \(\mu(p^k\mathbb Z_p)=p^{-k}\) for positive \(k\); the same finite-coset identity gives it for negative \(k\). Proposition 2.1 now gives (16) and the final assertion. \(\square\)

<a id="ha-lca-17-proposition-6-2"></a>
**Proposition 6.2 — Finite adèles, the solenoid, and Poisson summation.** Let
\[
 \mathbb A_f=\prod_p'(\mathbb Q_p;\mathbb Z_p),\qquad
 \mathbb A=\mathbb R\times\mathbb A_f,\qquad
 K_f=\prod_p\mathbb Z_p.
\]
The group \(\mathbb A_f\) belongs to \(\mathcal T\). The diagonal \(\mathbb Q\) is closed, discrete and cocompact in \(\mathbb A\), and
\[
 \mathbb A/\mathbb Q\cong\Sigma.                            \tag{17}
\]
Take \(dx=dx_\infty\,dx_f\), with \(dx_\infty\) Lebesgue measure and \(dx_f(K_f)=1\). The pairing
\[
 \Psi(x,y)=e^{-2\pi i x_\infty y_\infty}
                  \prod_p\psi_p(x_py_p)                   \tag{18}
\]
is a self-duality of \(\mathbb A\), with self-dual measure \(dx\), and \(\mathbb Q^\perp=\mathbb Q\). For
\[
 \mathcal S(\mathbb A)
 =\operatorname{span}\{g\otimes v:g\in\mathcal S(\mathbb R),
                                      \ v\in\mathcal S(\mathbb A_f)\},
\]
where \(\mathcal S(\mathbb R)\) denotes smooth functions rapidly decreasing with all derivatives, one has
\[
 \sum_{r\in\mathbb Q} f(x+r)
   =\sum_{r\in\mathbb Q}\widehat f(r)\Psi(x,r).              \tag{19}
\]
Both sums converge absolutely; the left sum converges uniformly on compact sets and the right sum uniformly everywhere.

**Proof.** The restricted-product construction, its local compactness, the compact open \(K_f\), the full Haar normalization, and the rational lattice with fundamental set
\([0,1)\times K_f\) are proved in [HA-LCA-11, Lemma 4.2](the-poisson-summation-formula.md#ha-lca-11-lemma-4-2). Its proof also gives the compact open neighbourhood base formed by restricting finitely many coordinates to \(p^{j_p}\mathbb Z_p\) and leaving the others integral. These clopen subgroups separate points, so \(\mathbb A_f\) is totally disconnected.

The finite-place pairing and its self-duality, and the full adelic pairing
\[
 B(x,y)=e^{2\pi i x_\infty y_\infty}
                         \prod_p\overline{\psi_p(x_py_p)}
\]
with \(K_f^\perp=K_f\), self-dual measure, and \(\mathbb Q^\perp=\mathbb Q\), are proved in [HA-LCA-11, Lemma 4.3](the-poisson-summation-formula.md#ha-lca-11-lemma-4-3). Thus \(\mathbb A_f\in\mathcal T\). Formula (18) is \(B(x,-y)\). Reflection \(y\mapsto-y\) is a continuous automorphism preserving Haar measure, as proved in [PRE-NONABELIAN-HAAR, Theorem 4.1](../prerequisites/src/nonabelian-haar-integration.md#ha-lca-pre-nonabelian-haar-theorem-4-1) in the abelian case. Consequently the conjugate pairing is also a topological self-duality with the same self-dual measure, and has the same annihilator of \(\mathbb Q\).

For (17), identify \(K_f\) with \(\mathbb Z_{\mathrm{prof}}\) by (12). Inclusion of \(\mathbb R\times K_f\) into \(\mathbb A\), followed by quotienting by \(\mathbb Q\), is onto because the fundamental set lies in this subgroup. Its kernel consists precisely of diagonal integers: the identity \(\mathbb Q\cap K_f=\mathbb Z\) is proved in HA-LCA-11, Lemma 4.2. It thus induces a continuous bijection \(\Sigma\to\mathbb A/\mathbb Q\). Compactness and the Hausdorff quotient topology make it a homeomorphism.

The test space here is exactly the one in [HA-LCA-11, Lemma 4.4](the-poisson-summation-formula.md#ha-lca-11-lemma-4-4); Theorem 3.2 identifies its finite-place space with \(\mathcal S(\mathbb A_f)\). That earlier lemma proves Fourier stability and all the absolute and uniform convergence assertions. [HA-LCA-11, Proposition 4.5](the-poisson-summation-formula.md#ha-lca-11-proposition-4-5) proves (19) for \(B\). Since \(\widehat f_{\Psi}(y)=\widehat f_B(-y)\), reindexing its absolutely convergent dual sum by \(r\mapsto-r\) gives exactly (19) for \(\Psi\). This proves all assertions with complete earlier arithmetic proofs and no external theorem in their place. \(\square\)

The sign conventions can be compared without changing the forward-transform rule (1):

| Pairing | Real basic character | Finite basic character at \(p\) | Forward kernel |
|---|---|---|---|
| \(B\), as in HA-LCA-11 | \(e^{2\pi it}\) | \(e^{-2\pi i\{x\}_p}\) | \(\overline{B(x,y)}\) |
| \(\Psi=\overline B\), as in (18) | \(e^{-2\pi it}\) | \(e^{2\pi i\{x\}_p}\) | \(B(x,y)\) |

Both global basic characters are trivial on the diagonal rationals. Their transforms differ by reflection of the frequency variable. On the single local field in (16) we use the explicitly stated positive \(\psi_p\).

## 7. Exercises with complete solutions

<a id="ha-lca-17-exercise-7-1"></a>
**Exercise 7.1 — Membership.** Decide which of
\(\mathbb Z,\mathbb R,\mathbb T,\mathbb Q/\mathbb Z,\mathbb Z_p,
\mathbb Q_p\times\mathbb Z,(\mathbb Z/2)^{\mathbb N}\), and
\(\bigoplus_{\mathbb N}\mathbb Z/2\) belong to \(\mathcal T\).

**Solution.** The answer, in that order, is
\[
 \text{no},\ \text{no},\ \text{no},\ \text{yes},\
 \text{yes},\ \text{no},\ \text{yes},\ \text{yes}.
\]
The discrete \(\mathbb Z\) has no finite subgroup containing \(1\); its compact subsets are finite, so Theorem 1.1 fails for \(\{1\}\). The real line and the circle are nontrivial connected groups; the connectedness of intervals follows from the intermediate value property [PRE-BANACH, Lemma 1.1](../prerequisites/src/banach-spectrum.md#ha-lca-pre-banach-lemma-1-1), and the circle is a continuous image of an interval. Thus neither is totally disconnected.

Every discrete torsion abelian group belongs to \(\mathcal T\): each compact subset is finite and generates a finite subgroup by the coefficient bound in Theorem 1.1. This proves the claims for \(\mathbb Q/\mathbb Z\) and the direct sum of copies of \(\mathbb Z/2\). Every profinite abelian group is compact totally disconnected, with discrete torsion dual by HA-LCA-12, Theorem 5.1 and Corollary 5.2, so it too belongs to \(\mathcal T\). This covers \(\mathbb Z_p\) and the indicated product of two-point groups, and also all finite abelian groups. Finally a compact subgroup of \(\mathbb Q_p\times\mathbb Z\) has finite subgroup image in \(\mathbb Z\), hence zero image. It cannot contain \((0,1)\), so Theorem 1.1 excludes that product. \(\square\)

<a id="ha-lca-17-exercise-7-2"></a>
**Exercise 7.2 — Stability.** Prove that \(\mathcal T\) is closed under closed subgroups, Hausdorff quotients, and finite products.

**Solution.** First a closed subgroup \(H\) of a totally disconnected LCA group inherits a compact open base \(H\cap K\). It is locally compact, since those intersections are compact neighbourhoods, and is totally disconnected since connected subsets remain connected in the ambient group.

A quotient \(G/H\) by a closed subgroup has compact open subgroups \(q(K)\), where \(K\) runs over the compact open subgroups of \(G\). They are compact by continuity and open because the quotient map is open. They form a base: given an identity neighbourhood \(U\) in the quotient, choose \(K\subset q^{-1}(U)\). Their clopen cosets separate points in the Hausdorff quotient, proving total disconnectedness there too. Local compactness and the quotient topology are the facts in HA-LCA-09, Lemma 6.0.

Now if \(G\in\mathcal T\), the duals satisfy
\(\widehat H\cong\widehat G/H^\perp\) and
\(\widehat{G/H}\cong H^\perp\), by HA-LCA-10, Theorem 2.1. The preceding subgroup and quotient arguments apply also to the totally disconnected \(\widehat G\). Thus both \(H\) and \(G/H\), and their duals, are totally disconnected. For a finite product, products of compact open subgroups give a compact open base. The finite product dual is the product of the duals by [HA-LCA-02, Proposition 4.1](characters-and-the-dual-group.md#ha-lca-02-proposition-4-1), so the same reasoning proves membership in \(\mathcal T\). \(\square\)

<a id="ha-lca-17-exercise-7-3"></a>
**Exercise 7.3 — A ball and its norm.** Compute the transform of \(1_{a+p^k\mathbb Z_p}\) for the pairing in (16), and verify Plancherel directly.

**Solution.** Translation by \(a\) contributes \(\psi_p(-ay)\). The character integral over \(p^k\mathbb Z_p\) is \(p^{-k}\) if \(y\in p^{-k}\mathbb Z_p\) and zero otherwise, giving exactly (16). The squared norm of the original indicator is \(p^{-k}\). The squared norm of its transform is
\[
 p^{-2k}\,\mu(p^{-k}\mathbb Z_p)
       =p^{-2k}p^k=p^{-k},
\]
as required for every integer \(k\). At \(a=0,k=0\) the same formula gives the self-transform of \(1_{\mathbb Z_p}\). \(\square\)

<a id="ha-lca-17-exercise-7-4"></a>
**Exercise 7.4 — The \(p\)-adic solenoid.** Identify the dual of the discrete group \(\mathbb Z[1/p]\).

**Solution.** The open union \(\mathbb Z[1/p]=\bigcup_{n\ge0}p^{-n}\mathbb Z\) and Proposition 4.1 give
\[
 \widehat{\mathbb Z[1/p]_d}
   \cong\varprojlim_{n\ge0}\mathbb R/p^n\mathbb Z
   \cong(\mathbb R\times\mathbb Z_p)/\{(k,k):k\in\mathbb Z\}.
                                                               \tag{20}
\]
For completeness, the second map sends \((t,z)\) to
\((t-z\bmod p^n)_n\), using the residue of \(z\) at each level. It is continuous and compatible; its kernel is the diagonal integers, because level \(n=0\) forces \(t\) integral and all the other levels then force \(z=t\). To prove surjectivity, choose \(t\) representing the level-zero circle coordinate. Subtracting the other coordinates from \(t\) gives a compatible family of integer residues modulo \(p^n\), hence \(z\in\mathbb Z_p\). The diagonal is closed and discrete by its real coordinate, and the quotient is compact since \([0,1]\times\mathbb Z_p\) maps onto it. The resulting continuous bijection onto the Hausdorff inverse limit is a homeomorphism. This proves both the group and topology in (20). \(\square\)

<a id="ha-lca-17-exercise-7-5"></a>
**Exercise 7.5 — A noncompact inverse limit.** Put \(D=\mathbb Q_p/\mathbb Z_p\), with the discrete topology. Prove
\[
 \theta:\mathbb Q_p\longrightarrow\varprojlim(D,\times p),
 \qquad \theta(x)_n=p^{-n}x+\mathbb Z_p\quad(n\ge0)          \tag{21}
\]
is a topological isomorphism. Deduce existence of a topological isomorphism
\(\mathbb Q_p\cong\widehat{\mathbb Q_p}\) without choosing a basic additive character on \(\mathbb Q_p\).

**Solution.** The coordinates in (21) are continuous homomorphisms and satisfy \(p\theta(x)_{n+1}=\theta(x)_n\). If all coordinates vanish, \(x\in\bigcap_np^n\mathbb Z_p=\{0\}\), so \(\theta\) is injective.

Given a compatible tuple \((d_n)\), choose representatives \(r_n\in\mathbb Q_p\). Compatibility says \(pr_{n+1}-r_n\in\mathbb Z_p\). The compact cosets
\[
 C_n=p^nr_n+p^n\mathbb Z_p
\]
are nonempty and nested, since \(p^{n+1}r_{n+1}-p^nr_n\in p^n\mathbb Z_p\). They all lie in the compact \(C_0\), so their intersection is nonempty by the finite intersection property. It has at most one point, because the difference of two such points lies in every \(p^n\mathbb Z_p\). Its unique point \(x\) satisfies \(\theta(x)_n=d_n\), proving surjectivity.

Continuity of all coordinates proves continuity into the inverse limit. Conversely the preimage of the condition that coordinate \(n\) be zero is \(p^n\mathbb Z_p\). These conditions form an identity-neighbourhood base in the inverse limit: a basic neighbourhood requires finitely many discrete coordinates to vanish, and vanishing of the largest one forces vanishing of all the earlier ones. Their preimages form an identity-neighbourhood base in \(\mathbb Q_p\). Thus the inverse of \(\theta\) is continuous.

To compute the dual independently of a basic character on the whole field, write
\(\mathbb Q_p=\bigcup_{n\ge0}p^{-n}\mathbb Z_p\).
Identify each stage with \(\mathbb Z_p\) by multiplication by \(p^n\). The inclusion of stage \(n\) in stage \(n+1\) then becomes multiplication by \(p\) on \(\mathbb Z_p\). Under the finite-residue dual identification
\(\widehat{\mathbb Z_p}\cong\mathbb Z[1/p]/\mathbb Z\cong D\)
of Proposition 4.3, pullback by this map is multiplication by \(p\) on \(D\). Proposition 4.1 gives
\(\widehat{\mathbb Q_p}\cong\varprojlim(D,\times p)\).
Combining with (21) gives the required topological isomorphism. Proposition 4.2 was not applied to the noncompact discrete stages \(D\); their inverse-limit topology was proved directly. \(\square\)

<a id="ha-lca-17-exercise-7-6"></a>
**Exercise 7.6 — Path components.** Prove that the neutral path component of \(\Sigma\) is the image of \(\mathbb R\times\{0\}\), and that there are uncountably many components.

**Solution.** Any path from zero is covered by finitely many of the interval-times-profinite charts of Theorem 5.1. Subdivide its parameter interval so each segment lies in one chart, using the Lebesgue-number argument. Lift the first segment to start at \((0,0)\). Its profinite coordinate is constant because the interval is connected and \(\mathbb Z_{\mathrm{prof}}\) is totally disconnected. At a junction, the next chart lift differs from the previous endpoint by a diagonal integer; translating the next lift by that integer makes the endpoints agree and keeps the constant profinite coordinate zero. Continuing over the finite subdivision proves that the endpoint lies in the real image. Conversely the path \(s\mapsto q(st,0)\) reaches every point in that image.

Translation identifies all other components with its cosets. The map taking the component of \(q(t,z)\) to \(z+\mathbb Z\) is well-defined and bijective, since two representatives differ by the real image exactly when their profinite coordinates differ by an integer. Lemma 5.0 identifies \(\mathbb Z_{\mathrm{prof}}\) with uncountable binary Cantor space. Each integer coset is countable, so countably many components would make \(\mathbb Z_{\mathrm{prof}}\) countable. Thus the component group is uncountable. \(\square\)