Arithmetic Satake compactifications and algebraic Drinfeld modular forms

Abstract

In this article, 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 Drinfeld modular forms over more general bases and the action of the (prime-to-residue characteristic and level) Hecke algebra. The construction also furnishes many algebraic Drinfeld modular forms obtained from the coefficients of the universal family that are also Hecke eigenforms. Among them, we obtain generalized Hasse invariants that are already defined on the arithmetic Satake compactification and not only its special fiber. We use these generalized Hasse invariants to study the geometry of the special fiber. We conjecture that our Satake compactification is Cohen–Macaulay. If this is the case, we 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 ${\mathrm{GL}}_r$.