Irrationality of finite logarithms in a congruence-class ad\`ele ring

Abstract

Finite logarithms can be defined in the ``poor man's ad\`{e}le ring" $\mathcal{A}$ using Fermat quotients modulo sufficiently large primes. This ring contains $\mathbb{Q}$ and outside the trivial cases, Matsusaka and Seki have shown that finite logarithms cannot take non-zero rational values in $\mathcal{A}$. Furthermore, a theorem of Silverman shows they are not zero, assuming the $abc$-conjecture. We extend these results to primes restricted to arithmetic progressions of the form $p\equiv 1\bmod m$ by relating Fermat quotients to values of cyclotomic polynomials and their logarithmic derivatives. A signed version of the same argument shows unconditionally that the square of a finite logarithm cannot take non-zero rational values in $\mathcal{A}$. As a further application, we show that, subject to the $abc$-conjecture, finite logarithms cannot be quadratic irrational elements of $\mathcal{A}$ in Rosen's theory of finite algebraic numbers.