Quantum modular forms, trigonometric products and quadratic irrationals

Bence Borda
TU Graz

Date(s) : 07/07/2020   iCal
11 h 00 min - 12 h 00 min

WEBINAIRE (lien: https://webconf.lal.cloud.math.cnrs.fr/b/pie-hmu-en9)

Résumé : Quantum invariants of hyperbolic knots have rich arithmetic structure. In this talk we consider an extension to the rationals of the so-called  Kashaev invariant of the figure eight knot, a quantum modular form in  the sense of Zagier. We find its asymptotics along the sequence of best  rational approximations to a given quadratic irrational, and show that  this asymptotics deviates from that of well approximable irrationals  recently established by Bettin and Drappeau. Our proof exploits the  close connection between the Kashaev invariant and Sudler’s  trigonometric product; in particular, we use a recent result of  Grepstad, Neumuller and Zafeiropoulos on the convergence of certain  normalized and shifted versions of Sudler’s products at quadratic irrationals.
Joint work with Christoph Aistleitner.

Bence Borda (TU Graz)

Site Sud, Luminy, TPR2, Salle de Séminaire 304-306 (3ème étage)


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