A short note on higher Mordell integrals

Males, Joshua

Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 2, pp. 563-573.

Classical mock modular and quantum modular forms are known to have an intimate relationship with Mordell integrals thanks to Zwegers groundbreaking Ph.D. thesis. More recently, generalisations of mock/quantum modular forms to so-called “higher depth” versions have been intensively studied. In essence, a mock/quantum modular form of depth $d$ is such that the error of modularity transforms as another mock/quantum modular form of depth $d-1$. In this short note we use techniques of Bringmann, Kaszian, and Milas to show that the double Eichler integrals of a family of depth two quantum modular forms of weight one previously studied by the author can be related to certain “higher” Mordell integrals, meaning it may be written as a certain double integral, à la Zwegers.

On sait que les formes modulaires fausses classiques et les formes modulaires quantiques sont intimement liées aux intégrales de Mordell grâce à la thèse de doctorat révolutionnaire de Zwegers. Plus récemment, certaines généralisations des formes modulaires fausses/quantiques, appelées formes de profondeur supérieure (« higher depth »), ont été étudiées de manière intensive. En gros, une forme modulaire fausse/quantique de profondeur $d$ est celle dont l’erreur de modularité se transforme comme une forme modulaire fausse/quantique de profondeur $d-1$. Dans cette courte note, nous utilisons des techniques de Bringmann, Kaszian et Milas pour montrer que les intégrales doubles d’Eichler d’une famille de formes modulaires quantiques de profondeur deux et de poids un, précédemment étudiées par l’auteur, peuvent être reliées à certaines intégrales de Mordell supérieures, ce qui signifie qu’elles peuvent être écrites comme une certaine intégrale double, à la Zwegers.

Classification: 11F12
Keywords: Quantum modular forms, higher Mordell integrals

[1] Alexandrov, Sergei; Banerjee, Sibasish; Manschot, Jan; Pioline, Boris Indefinite theta series and generalized error functions, Sel. Math., New Ser., Volume 24 (2018) no. 5, pp. 3927-3972 | DOI | MR | Zbl

[2] Bringmann, Kathrin; Folsom, Amanda; Rhoades, Robert C. Unimodal sequences and “strange” functions: a family of quantum modular forms, Pac. J. Math., Volume 274 (2015) no. 1, pp. 1-25 | DOI | MR | Zbl

[3] Bringmann, Kathrin; Kaszian, Jonas; Milas, Antun Higher depth quantum modular forms, multiple Eichler integrals, and ${\mathrm{𝔰𝔩}}_3$ false theta functions, Res. Math. Sci., Volume 6 (2019) no. 2, 20, 41 pages | MR | Zbl

[4] Bringmann, Kathrin; Kaszian, Jonas; Milas, Antun Vector-valued higher depth quantum modular forms and higher Mordell integrals, J. Math. Anal. Appl., Volume 480 (2019) no. 2, 123397, 22 pages | MR | Zbl

[5] Bringmann, Kathrin; Rolen, Larry Half-integral weight Eichler integrals and quantum modular forms, J. Number Theory, Volume 161 (2016), pp. 240-254 | DOI | MR | Zbl

[6] Bryson, Jennifer; Ono, Ken; Pitman, Sarah; Rhoades, Robert C. Unimodal sequences and quantum and mock modular forms, Proc. Natl. Acad. Sci. USA, Volume 109 (2012) no. 40, pp. 16063-16067 | DOI | MR

[7] Folsom, Amanda; Ki, Caleb; Truong Vu, Yen Nhi; Yang, Bowen “Strange” combinatorial quantum modular forms, J. Number Theory, Volume 170 (2017), pp. 315-346 | DOI | MR | Zbl

[8] Hikami, Kazuhiro; Kirillov, Anatol N. Torus knot and minimal model, Phys. Lett., B, Volume 575 (2003) no. 3, pp. 3-4 | MR | Zbl

[9] Hikami, Kazuhiro; Lovejoy, Jeremy Torus knots and quantum modular forms, Res. Math. Sci., Volume 2 (2015) no. 1, 2, 15 pages | MR | Zbl

[10] Kronecker, Leopold Bemerkungen über die Darstellung von Reihen durch Integrale, J. Reine Angew. Math., Volume 105 (1889), pp. 157-159 | DOI | Zbl

[11] Kronecker, Leopold Summirung der Gausschen Reihen, J. Reine Angew. Math., Volume 105 (1889), pp. 267-268 | DOI | MR

[12] Males, Joshua A family of vector-valued quantum modular forms of depth two, Int. J. Number Theory, Volume 16 (2020) no. 1, pp. 29-64 | DOI | MR | Zbl

[13] Mordell, Louis The value of the definite integral ${\int }_{-\infty }^{\infty }\frac{e^{a{t}^2+bt}}{e^{c}t+d}\mathrm{d}t$, Q. J. Math., Volume 68 (1920), pp. 329-342

[14] Mordell, Louis The definite integral ${\int }_{-\infty }^{\infty }\frac{e^{a{x}^2+bx}}{e^{a}x+d}da$ and the analytic theory of numbersand the analytic theory of numbers, Acta Math., Volume 61 (1933), pp. 323-360 | DOI

[15] Ono, Ken et al. Unearthing the visions of a master: harmonic Maass forms and number theory, Current developments in mathematics, 2008, International Press, 2009, pp. 347-454 | Zbl

[16] Rolen, Larry; Schneider, Robert P. A “strange” vector-valued quantum modular form, Arch. Math., Volume 101 (2013) no. 1, pp. 43-52 | DOI | MR | Zbl

[17] Siegel, Carl L. Über Riemanns Nachlass zur analytischen Zahlentheorie, Quell. Stud. Gesch. Math. B, Volume 2 (1932), pp. 45-80 | Zbl

[18] Zagier, Don Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology, Volume 40 (2001) no. 5, pp. 945-960 | DOI | MR | Zbl

[19] Zagier, Don Quantum modular forms, Quanta of maths (Clay Mathematics Proceedings), Volume 11, American Mathematical Society, 2010, pp. 659-675 | MR | Zbl

[20] Zwegers, Sander Mock theta functions, Ph. D. Thesis, Universiteit Utrecht (The Netherland) (2002)