Irrationality of finite logarithms in a congruence-class adèle ring

Technical Summary: Irrationality of Finite Logarithms in a Congruence-Class Adèle Ring

Problem Statement
The paper investigates the arithmetic properties of "finite logarithms" within the "poor man's adèle ring" $\mathcal{A}$, defined as the restricted product of $\mathbb{Z}/p\mathbb{Z}$ over all primes $p$. Elements in this ring are sequences $(t_p)_{p \in \mathbb{P}}$ where $t_p \in \mathbb{Q}$, identified modulo $p$ for sufficiently large primes. The finite logarithm function $\log_{\mathcal{A}}: \mathbb{Q}^\times \to \mathcal{A}$ is defined via Fermat quotients:
$$\log_{\mathcal{A}}(\alpha) = (q_p(\alpha) \pmod p)_{p \in \mathbb{P}}, \quad \text{where } q_p(\alpha) = \frac{\alpha^{p-1}-1}{p}.$$
Previous work by Matsusaka and Seki established that for $\alpha \in \mathbb{Q}^\times \setminus \{\pm 1\}$, $\log_{\mathcal{A}}(\alpha)$ is never a non-zero rational element in $\mathcal{A}$. Furthermore, Silverman showed (conditional on the $abc$-conjecture) that $\log_{\mathcal{A}}(\alpha) \neq 0$.

The central problem addressed in this paper is to extend these irrationality and non-vanishing results to restricted sets of primes. Specifically, the author considers primes restricted to arithmetic progressions of the form $p \equiv 1 \pmod m$ for a fixed integer $m \ge 1$. The goal is to determine if $\log_{\mathcal{A}}(\alpha)$ can take rational values or vanish within the restricted ring $\mathcal{A}(m)$, and to analyze the "recurrence degree" of these logarithms, specifically whether they can be quadratic irrational elements.

Methodology
The paper employs a combination of cyclotomic polynomial theory and classical irrationality proof techniques adapted to the adèle setting.

1. Cyclotomic Constructions: The author relates Fermat quotients to values of cyclotomic polynomials $\Phi_n(u, v)$ and their partial derivatives. A key identity utilized is the homogenized form $\Phi_n(X^p, Y^p) = \Phi_n(X, Y)\Phi_{np}(X, Y)$.
2. Congruence Analysis: By analyzing the prime factors of $\Phi_n(u, v)$, the paper establishes that if a prime $p$ divides $\Phi_n(u, v)$ (with $p \nmid n$), then $p \equiv 1 \pmod n$. This property is crucial for restricting the analysis to the congruence class $p \equiv 1 \pmod m$.
3. Contradiction via Vanishing Sequences: To prove irrationality (Theorem 1.2(i)), the author assumes $\log_{\mathcal{A}(m)}(\alpha)$ is rational. This assumption allows the construction of a sequence of non-zero integers derived from the values of $\Phi_{m\ell}(u, v)$ and their derivatives. The proof demonstrates that this sequence, when normalized, tends to zero as the auxiliary prime $\ell \to \infty$, yielding a contradiction.
4. Application of the $abc$-Conjecture: To prove non-vanishing (Theorem 1.2(ii)), the paper adapts Silverman's method. It utilizes the $abc$-conjecture to bound the "powerful part" of $u^n - v^n$. By showing that the cyclotomic values grow too rapidly to be "powerful" (square-full) under the assumption that the logarithm vanishes, a contradiction is reached.
5. Recurrence Degree Classification: The paper defines the recurrence degree $\deg_{\mathcal{A}}(t)$ as the minimal order of a linear recurrence with rational coefficients satisfied by the sequence representing $t \in \mathcal{A}$. It classifies elements of degree $\le 2$ using quadratic Dirichlet characters.

Key Contributions and Results

• Extension to Congruence Classes (Theorem 1.2): The paper proves that for any $\alpha \in \mathbb{Q}^\times \setminus \{\pm 1\}$ and $m \ge 1$:

  • (i) $\log_{\mathcal{A}(m)}(\alpha)$ is not a non-zero rational number in the restricted ring $\mathcal{A}(m)$.
  • (ii) Assuming the $abc$-conjecture, $\log_{\mathcal{A}(m)}(\alpha) \neq 0$ in $\mathcal{A}(m)$.
  • Note: The restriction $p \equiv 1 \pmod m$ is intrinsic to the cyclotomic method used; the author notes that forcing other residue classes $p \equiv b \pmod m$ (where $b \not\equiv 1$) is not amenable to this specific approach.

• Quadratic Irrationality (Theorem 1.3): The paper investigates whether finite logarithms can be "quadratic" in the sense of having a recurrence degree $\le 2$.

  • It is shown that if $\deg_{\mathcal{A}}(\log_{\mathcal{A}}(\alpha)) \le 2$, then $\log_{\mathcal{A}}(\alpha)$ must take the specific form $r(1 - (\frac{D}{p}))_p$ for some $r \in \mathbb{Q}$ and square-free integer $D$.
  • Crucially, assuming the $abc$-conjecture, the paper proves that $\deg_{\mathcal{A}}(\log_{\mathcal{A}}(\alpha)) > 2$. This implies that finite logarithms cannot be quadratic irrational elements in $\mathcal{A}$.

• Generalization to Abelian Extensions (Remark 4.2): The argument is extended to suggest that if the splitting field of the characteristic polynomial of a recurrence is abelian, the logarithm cannot lie in the image of that recurrence (again, assuming the $abc$-conjecture).

Significance and Claims
The paper claims to extend the known irrationality results of Matsusaka, Seki, and Silverman from the full adèle ring to rings restricted to specific arithmetic progressions. This is significant because it demonstrates the robustness of the irrationality of finite logarithms even when the underlying prime set is thinned out to a density of $1/\phi(m)$.

Furthermore, the work connects the arithmetic of Fermat quotients to the theory of finite algebraic numbers (in the sense of Rosen). By proving that $\log_{\mathcal{A}}(\alpha)$ cannot be a quadratic element (and more generally, cannot lie in the image of recurrences with abelian splitting fields), the paper provides evidence supporting the broader conjecture that finite logarithms are "naively transcendental" (i.e., not algebraic in the sense of satisfying linear recurrences). The author modestly notes that the non-abelian case remains open due to the lack of a straightforward cyclotomic approach for primes in non-identity residue classes.