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January 22, 2018 | 2:45 p.m. - 4:00 p.m.
Category: Lecture
Location: Education, College of #179 | Map
5425 Gullen Mall
Detroit, MI 48202
Cost: Free

Speaker:  Luca Candelori, University of Hawaii

Title: The search for Riemann surfaces with complex multiplication

Abstract:  Despite more than a century of intense efforts, the relation between a Riemann surface and its Jacobian variety is still shrouded in mystery. A prime example of this unsatisfying state of affairs is the theory of CM Riemann surfaces, those Riemann surfaces whose Jacobian is an abelian variety with complex multiplication (“CM” in short). While we have a good understanding of the theory of CM abelian varieties, there is no systematic way to produce CM Riemann surfaces: the only known examples come from special surfaces with large automorphism groups, such as Fermat curves, and yet a generic Riemann surface should have trivial automorphism group. 

In this talk we unveil for the first time a new method expected to produce explicitly all CM Riemann surfaces. More precisely, we give a new cohomological criterion to detect CM and a new algorithm to compute the period matrix of a CM Riemann surface. This period matrix is known to have algebraic entries, which we can compute exactly without the need of numerical approximation. Our algorithm is based on the theory of dessins d’enfants, the  uniformization theory of Riemann surfaces and mostly on  giving a fresh new look at the glue binding all these beautiful ideas together: the theory of modular forms.

For more information about this event, please contact Department of Mathematics at 3135772479 or math@wayne.edu.