Irrationality and transcendence questions in the "poor man's adèle ring"

2025/05/14 by Luca, Florian, Zudilin, Wadim · 2 citations
#11A41 #11B39 #11B68 #11J72 #11J81 (primary) #16U10 (secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.09775

Abstract

We discuss arithmetic questions related to the "poor man's adèle ring" \mathcal A whose elements are encoded by sequences (t_p)_p indexed by prime numbers, with each t_p viewed as a residue in \mathbb Z/p\mathbb Z. Our main theorem is about the \mathcal A-transcendence of the element (F_p(q))_p, where F_n(q) (Schur's q-Fibonacci numbers) are the (1,1)-entries of 2×2-matrices (\beginmatrix 1 amp; 1
1 amp; 0 \endmatrix) (\beginmatrix 1 amp; 1
q amp; 0 \endmatrix) (\beginmatrix 1 amp; 1
q² amp; 0 \endmatrix) ⋯ (\beginmatrix 1 amp; 1
q^n-2 amp; 0 \endmatrix) and q>1 is an integer. This result was previously known for q>1 square free under the GRH.

Cited by

• Irrationality of finite logarithms in a congruence-class adèle ring
• Poor man's transcendence for Frobenius traces of elliptic curves

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