Counting roots of unity on the graphs of Laurent series over non-Archimedean local fields

2025/11/30 by Christoph Pütz, Pütz, Christoph
Computer Science · Mathematics · #11D88 #11G99 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2512.00982

openalex publication_date 2025/11/30 · openalex created_date 2025/12/03 · openalex updated_date 2026/07/28

Abstract

We completely classify Laurent series converging on the unit circle over a non-Archimedean local field (of any characteristic) that map infinitely many roots of unity to roots of unity. For a given Laurent series f over a field of positive characteristic with residue field \mathbbF_q, we prove effective bounds for the number of possible roots of unity in terms of the number of zeroes of the auxilliary function f(x^q)-f(x)^q on the unit circle. In characteristic 0 our bound is still effective but also depends on the ramification degree of the base field over ℚ_p as well as the size of the coefficients of f. This has applications to the Manin-Mumford conjecture in \mathbbG_m². In characteristic 0, this work builds upon a pigeon-hole based method by Schmidt.

Citations

• On p-adic versions of the Manin-Mumford Conjecture
• Rational values of transcendental functions and arithmetic dynamics
• Courbes sur une vari�t� ab�lienne et points de torsion

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