Integer-valued polynomials and p-adic Fourier theory

2025/02/25 by Berger, Laurent, Sprang, Johannes
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.18053

Abstract

The goal of this paper is to give a numerical criterion for an open question in p-adic Fourier theory. Let F be a finite extension of Q_p. Schneider and Teitelbaum defined and studied the character variety \mathfrakX, which is a rigid analytic curve over F that parameterizes the set of locally F-analytic characters λ: (o_F,+) → (C_p^×,×). Determining the structure of the ring Λ_F(\mathfrakX) of bounded-by-one functions on \mathfrakX defined over F seems like a difficult question. Using the Katz isomorphism, we prove that if F= Q_p², then Λ_F(\mathfrakX) = o_F [ [o_F] ] if and only if the o_F-module of integer-valued polynomials on o_F is generated by a certain explicit set. Some computations in SageMath indicate that this seems to be the case.

Related

• Bounded functions on the character variety
• p-adic Fourier Theory
• p-adic Fourier theory in families
• Divided powers in Chow rings and integral Fourier transforms
• Another look at p-adic Fourier-theory