Lecture Room B, 4th Floor, The 3rd General Building, NTHU

(清華大學綜合三館 4樓B演講室)

Arithmetic Satake Compactifications and Algebraic Drinfeld Modular Forms

Chia-Fu Yu (Academia Sinica)

Abstract:

In this joint work with Urs Hartl we construct the arithmetic Satake compactification of the Drinfeld moduli schemes of arbitrary rank over the ring of integers of any global function field away from the level structure, and show that the universal family extends uniquely to a generalized Drinfeld module over the compactification. Using these and functorial properties, we define algebraic Drinfeldmodular forms over more general bases and the action of the (prime-to-residue characteristic) Hecke algebra. The construction also furnishes many algebraic Drinfeld modular forms obtained from the coefficients of the universal family which are also Hecke eigenforms. Among them we obtain generalized Hasse invariants which already live on the arithmetic Satake compactification. We use these generalized Hasse invariants to study the geometry of the special fiber and to establish the Jacquet-Langlands correspondence (mod v) between Hecke eigensystems of rank r Drinfeld modular forms and those of algebraic modular forms (in the sense of Gross) attached to a compact inner form of GL_r.