Defining the Primal Manifold as an Adele Quotient Space

A mash up of analytic number theory, algebraic number theory, and topological group theory, primarily referencing the framework established by Iwasawa and Tate.

Assume the "Primal Manifold" (denote it $\mathcal{P}$) is conjectured to be the fundamental geometric space whose properties (e.g., its spectrum, cohomology, or character space) relate to the non-trivial zeros of the Riemann zeta function $\zeta(s)$, then defining it as an adele quotient space has a natural candidate within the standard theory.

The most logical construction would identify $\mathcal{P}$ with the compact part of the Idele Class Group of $\mathbb{Q}$. Here is a formal mathematical outline of such a definition:

1. Building Blocks: Adeles and Ideles

To define the quotient space, first establish the foundational spaces for the number field $K = \mathbb{Q}$.

• Completions: For each prime $p$, the $p$-adic completion $\mathbb{Q}_p$ (with its non-archimedean absolute value $|\cdot|_p$ and maximal compact subring $\mathbb{Z}_p$). At the infinite place, we have the real completion $\mathbb{Q}_\infty = \mathbb{R}$ (with the standard absolute value $|\cdot|_\infty$).
• The Adele Ring ($A_{\mathbb{Q}}$): The adele ring is the restricted direct product of all completions: $$A_{\mathbb{Q}} := \prod_{v}' \mathbb{Q}_v = \mathbb{R} \times \prod_{p}' \mathbb{Q}_p$$ An element $a = (a_\infty, a_2, a_3, \dots) \in A_{\mathbb{Q}}$ is a sequence where $a_v \in \mathbb{Q}_v$, and for all but finitely many primes $p$, $a_p \in \mathbb{Z}_p$. This makes $A_{\mathbb{Q}}$ a locally compact topological ring.
• The Idele Group ($J_{\mathbb{Q}}$): The idele group is the group of invertible elements of the adele ring, $J_{\mathbb{Q}} = A_{\mathbb{Q}}^\times$. Its also a restricted product: $$J_{\mathbb{Q}} := \prod_{v}' \mathbb{Q}_v^\times = \mathbb{R}^\times \times \prod_{p}' \mathbb{Q}_p^\times$$ An element $j = (j_\infty, j_2, j_3, \dots) \in J_{\mathbb{Q}}$ has $j_p \in \mathbb{Z}_p^\times$ (the $p$-adic units) for all but finitely many primes $p$.
• Principal Subgroup: The field $\mathbb{Q}$ embeds diagonally into $A_{\mathbb{Q}}$ as a discrete, co-compact subgroup. Similarly, the multiplicative group $\mathbb{Q}^\times$ embeds diagonally into $J_{\mathbb{Q}}$ as a discrete subgroup.

2. Formal Definition of the Primal Manifold ($\mathcal{P}$)

The central object in Tate's thesis for studying the $\zeta$-function is the Idele Class Group, $C_{\mathbb{Q}} = J_{\mathbb{Q}} / \mathbb{Q}^\times$. This space is not compact. However, it contains a canonical compact subgroup that is a candidate for the Primal Manifold:

The Idele Norm

Defining the global norm (or modulus) on the idele group, $|\cdot|: J_{\mathbb{Q}} \to \mathbb{R}_+^\times$, as: $$|j| = \prod_v |j_v|_v = |j_\infty|_\infty \prod_p |j_p|_p$$ This is a continuous homomorphism. By the Artin Product Formula, this map is trivial on the principal ideles: for any $q \in \mathbb{Q}^\times$, $|q| = 1$.

Definition

Because the norm is trivial on $\mathbb{Q}^\times$, it descends to a map on the idele class group, $|\cdot|: C_{\mathbb{Q}} \to \mathbb{R}_+^\times$.

Formally defining the Primal Manifold $\mathcal{P}$ as the kernel of this norm map; that is, $\mathcal{P}$ is the norm-1 subgroup of the idele class group.

This space $\mathcal{P}$ is a compact (by Gelfand-Raikov), connected (topologically non-trivial), and infinitely-divisible abelian group.

3. Equivalent Formulations (The "Debate")

This centers on the different but isomorphic ways to describe this single object, $\mathcal{P}$, each highlighting a different aspect of its structure (topological, arithmetic, or group-theoretic).

Formulation A: The Quotient of Norm-1 Ideles

This is the most direct definition. Let $J_{\mathbb{Q}}^1$ be the kernel of the norm map on the idele group itself: $J_{\mathbb{Q}}^1 = \{ j \in J_{\mathbb{Q}} : |j|=1 \}$. Since $\mathbb{Q}^\times \subset J_{\mathbb{Q}}^1$ (by the product formula), our definition is equivalent to: $$\mathcal{P} = J_{\mathbb{Q}}^1 / \mathbb{Q}^\times$$ This defines $\mathcal{P}$ as the quotient of the (non-compact) norm-1 ideles by the principal ideles.

Formulation B: The Profinite Arithmetic Definition

Global class field theory provides a fundamental isomorphism for the full idele class group: $$C_{\mathbb{Q}} = J_{\mathbb{Q}} / \mathbb{Q}^\times \cong \mathbb{R}_+^\times \times \hat{\mathbb{Z}}^\times$$ where $\hat{\mathbb{Z}}^\times = \prod_p \mathbb{Z}_p^\times$ is the group of profinite units (i.e., the units of the ring of profinite integers $\hat{\mathbb{Z}}$).

In this decomposition, the norm map $|(t, u)| = t$ (where $t \in \mathbb{R}_+^\times, u \in \hat{\mathbb{Z}}^\times$) is simply the projection onto the first factor. The kernel is therefore $\{1\} \times \hat{\mathbb{Z}}^\times$. This gives the profound isomorphism:

This formulation defines $\mathcal{P}$ as a profinite group. It is compact, Hausdorff, and totally disconnected (a Cantor set). From an arithmetic perspective, this is $\text{Gal}(\mathbb{Q}^{ab} / \mathbb{Q})$'s maximal compact subgroup, or more simply, $\text{Gal}(\mathbb{Q}(\mu_\infty)/\mathbb{Q})$.

Formulation C: The Additive Analogue (A Solenoid)

A different, but related, adele quotient is the additive one: $$\mathbb{S} := A_{\mathbb{Q}} / \mathbb{Q}$$ This is the adele solenoid. It is a compact and connected group (and the Pontryagin dual of the discrete group $\mathbb{Q}$).

Now the question is if $\mathcal{P}$ should be the multiplicative space $\mathcal{P} = C_{\mathbb{Q}}^1$ (which is $J_{\mathbb{Q}}^1 / \mathbb{Q}^\times \cong \hat{\mathbb{Z}}^\times$) or this additive space $\mathbb{S}$. Given that the $\zeta$-function is built from a multiplicative zeta-integral over $J_{\mathbb{Q}}$ (as in Tate's thesis), the multiplicative candidate $\mathcal{P} = C_{\mathbb{Q}}^1$ is the more direct choice.

Summary: Connection to the Riemann Hypothesis

The characters of $C_{\mathbb{Q}}$ are the "Grössencharaktere" of Hecke. Any character $\chi: C_{\mathbb{Q}} \to \mathbb{C}^\times$ can be uniquely written as: $$\chi(j) = \omega(j) \cdot |j|^s$$ for some $s \in \mathbb{C}$ and some character $\omega$ that is trivial on the connected component $\mathbb{R}_+^\times$. This $\omega$ is a character of our $\mathcal{P} = C_{\mathbb{Q}}^1$.

The Riemann Hypothesis is a statement about the zeros of L-functions, which are Mellin transforms of theta functions associated with these characters. Defining $\mathcal{P}$ as the compact adele quotient $C_{\mathbb{Q}}^1$ provides the fundamental "unitary" space $\mathcal{P} \cong \hat{\mathbb{Z}}^\times$ on which the "arithmetic" part $\omega$ of the zeta-character lives. The RH from this perspective, relates to the spectrum of the operator (the Laplacian) on the non-compact part of this adelic space.

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