Quantum modularity of partial theta series with periodic coefficients

被引：9

作者：

Goswami, Ankush ^{[2
]}

Osburn, Robert ^{[1
]}

机构：

[1] Univ Coll Dublin, Sch Math & Stat, Dublin 4, Ireland

[2] Johannes Kepler Univ Linz, Res Inst Symbol Computat RISC, A-4040 Linz, Austria

来源：

FORUM MATHEMATICUM | 2021年 / 33卷 / 02期

基金：

奥地利科学基金会;

关键词：

Quantum modular forms; partial theta series; periodic functions; Kontsevich-Zagier series; torus knots; INVARIANTS; FORMS;

D O I：

10.1515/forum-2020-0201

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich-Zagier series F-t (q) which matches (at a root of unity) the colored Jones polynomial for the family of torus knots T(3, 2(t)), t >= 2, is a weight 3/2 quantum modular form. This generalizes Zagier's result on the quantum modularity for the "strange" series F(q).

引用

页码：451 / 463

页数：13

共 30 条

[1]

Bettin S., 2019, PREPRINT

[2] Generalized Fishburn numbers and torus knots [J].

Bijaoui, Colin ;

Boden, Hans U. ;

Myers, Beckham ;

Osburn, Robert ;

Rushworth, William ;

Tronsgard, Aaron ;

Zhou, Shaoyang .

JOURNAL OF COMBINATORIAL THEORY SERIES A, 2021, 178

[3]

Bringmann K., 2017, Harmonic Maass Forms and Mock Modular Forms: Theory and Applications, V64

[4]

Bringmann K., 2019, Res. Math. Sci., V6

[5] Quantum modular forms and plumbing graphs of 3-manifolds [J].

Bringmann, Kathrin ;

Mahlburg, Karl ;

Milas, Antun .

JOURNAL OF COMBINATORIAL THEORY SERIES A, 2020, 170

[6] Half-integral weight Eichler integrals and quantum modular forms [J].

Bringmann, Kathrin ;

Rolen, Larry .

JOURNAL OF NUMBER THEORY, 2016, 161 :240-254

[7] W-Algebras, False Theta Functions and Quantum Modular Forms, I [J].

Bringmann, Kathrin ;

Milas, Antun .

INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2015, 2015 (21) :11351-11387

[8] Negative index Jacobi forms and quantum modular forms [J].

Bringmann, Kathrin ;

Creutzig, Thomas ;

Rolen, Larry .

RESEARCH IN THE MATHEMATICAL SCIENCES, 2014, 1 (01)

[9]

Bruggeman R, 2007, Conference on L-Functions, P1

[10] Three-manifold quantum invariants and mock theta functions [J].

Cheng, Miranda C. N. ;

Ferrari, Francesca ;

Sgroi, Gabriele .

PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 2020, 378 (2163)

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