On quantum modular forms of non-zero weights
Résumé
We study functions $f$ on $\mathbb Q$ which statisfy a ``quantum modularity'' relation of the shape $$ f(x+1)=f(x), \qquad f(x) - |x|^{-k} f(-1/x) = h(x) $$ where $h:\mathbb R_{\neq 0} \to \mathbb C$ is a function satisfying various regularity conditions. We study the case $\Re(k)\neq 0$. 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 $\Re(k)\neq0$ 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\leq a
Domaines
Théorie des nombres [math.NT]
Origine : Fichiers produits par l'(les) auteur(s)
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https://hal.science/hal-04052954
Soumis le : jeudi 30 mars 2023-22:05:54
Dernière modification le : jeudi 18 avril 2024-16:45:01
Citer
Sandro Bettin, Sary Drappeau. On quantum modular forms of non-zero weights. 2022. ⟨hal-04052954⟩
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