Seminar

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In a seminal Crelle article whose quadragennial anniversary will be celebrated next year, Gross and Zagier studied the factorisation of differences of singular moduli (values of the modular j-function at CM points). More recently, Jan Vonk and I developped  an analogous picture in which quadratic imaginary points on the Poincaré upper half plane are replaced by real quadratic points on the Drinfeld upper half plane. Our proof of the algebraicity and factorisation of the resulting p-adic invariants is modeled on the ``analytic proof” in the article of Gross and Zagier. We will compare and contrast the two approaches, hinting at the existence of  ``arithmetic theta liftings” in settings where  special cycles on Shimura varieties are ostensibly unavailable, and the breach must be filled by the (weaker) deformation theory of p-adic modular forms and their associated Galois representations.

ES Program Research Seminar: "Differences Of Singular Moduli: A Tale Of Two Calculations"