Time：10:00-11:00, Thursday, November 27 2025
Venue：E4-233
Speaker：Luca Demangos, Xi'an Jiaotong University
Title：Quantum Drinfeld modules and the Real Multiplication program
Abstract：The classical theory of Complex Multiplication was developed in the XIX century by Fueter and Weber, to arithmetically describe the abelian closure of any quadratic imaginary number field K. More specifically, to the rank 2 lattice generated by 1 and a primitive generator of K over Q, it is attached an analytic isomorphism with a projective curve, endowing it with an algebraic group operation, induced by the ordinary sum on C, which makes it a CM elliptic curve. Its torsion points are precisely the generators of the abelian closure of K over the field of definition, generated in turn over K by the value taken by the j invariant on the corresponding point of the moduli space. Any naif tentative to replicate this construction on real quadratic number fields (the Real Multiplication program) fails, as the rank 2 group generated by 1 and the primitive generator of K is now dense in R, and no arithmetic theory can be produced on it. We propose an approach based on diophantine approximation, which we have tested on global function fields arena. Our notion of quantum Drinfeld module provides the analog of the theory of Fueter-Weber for real quadratic global function fields. The work is in collaboration with prof. T. M. Gendron (Universidad Nacional Autonoma de Mexico).
