分野別セミナー

For a fundamental discriminant D<0, "singular moduli" means the value of a modular function at a CM point with discriminant D. Thanks to results of Zagier (for the case of level one), and Bruinier and Funke (for the case of general level), the generating function of modular traces of singular moduli for a fixed modular function is a mock forms of weight 3/2. For a fundamental discriminant D>0, Duke, Imamoglu, and Toth gave the definition of singular moduli of a modular function by using the integral of the modular function on a certain geodesic of a modular curve. They showed that modular traces of singular moduli (D>0) for a fixed modular function is a mock forms of weight 1/2.

Based on this progress on modular traces of singular moduli, in this talk I will talk about the following question: for a fixed modular function f, to find arithmetic connections between traces of singular moduli of f (d > 0) and those of f (d < 0) . To introduce our results with Subong Lim on this question, first I will review basic notations of mock modular forms and results on modularity of traces of singular moduli. Next, I will announce our results and then give a brief sketch of the proof of the results.