A short note on higher Mordell integrals

Males, Joshua

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

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.

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.

Classification : 11F12
Mots-clés : Quantum modular forms, higher Mordell integrals

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