On quantum modular forms of non-zero weights

2022/10/14 by Sandro Bettin, Bettin, Sandro, Sary Drappeau +1
Mathematics · #11A05 #11F03 (Primary) #11F20 #11F67 #11F99 (Secondary) #11K50 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2210.07854

openalex publication_date 2022/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We study functions f on \mathbb Q which statisfy a ``quantum modularity'' relation of the shape f(x+1)=f(x), f(x) - |x|^-k f(-1/x) = h(x) where h:\mathbb R_{≠ 0} → \mathbb C is a function satisfying various regularity conditions. We study the case \Re(k)≠ 0. We prove the existence of a limiting function f^* which extends continuously f to \mathbb R in some sense. This means in particular that in the \Re(k)≠0 case the quantum modular form itself has to have at least a certain level of regularity. We deduce that the values \f(a/q), 1≤ a

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