Proportionality and the Arithmetic Volumes of Shimura Varieties and the Moduli of Drinfeld Shtukas

Explore the deep connections between arithmetic geometry and special values of L-functions in this advanced mathematical lecture that extends Hirzebruch's proportionality theorem to new contexts. Learn how the volume of locally symmetric spaces relates to products of special values of zeta functions, and discover how this classical result generalizes to function fields where locally symmetric spaces are replaced by moduli spaces of Drinfeld Shtukas with multiple legs. Examine the speaker's proof of analogous results where special values of zeta functions are replaced by linear combinations of their derivatives of various orders. Investigate a new conjecture formulated over number fields that relates arithmetic volumes, defined by Arakelov degrees of certain Hermitian metrized line bundles, to special values of derivatives of suitable Artin L-functions, along with proofs in several new cases. Gain insights into cutting-edge research in arithmetic geometry, modular forms, and the arithmetic of Shimura varieties through this collaborative work with Tony Feng and Zhiwei Yun, presented by Wei Zhang from MIT at the Institut des Hautes Etudes Scientifiques.