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Arithmetic Satake compactifications and algebraic Drinfeld modular forms

Urs Hartl, Chia-Fu Yu

TL;DR

This work develops an arithmetic framework for Drinfeld modular varieties by constructing the arithmetic Satake compactification of rank $r$ Drinfeld moduli over $A_{(v)}$ and showing the universal Drinfeld module extends to a generalized Drinfeld module on the compactification. It defines algebraic Drinfeld modular forms over general bases through global sections of the ample line bundle $\omega_K$, and analyzes their Hecke actions, including eigenforms modulo $v$. The paper introduces generalized Hasse invariants to stratify the special fiber, connects supersingular Drinfeld modules to algebraic modular forms on inner forms of $\mathrm{GL}_r$ via a mod $v$ Jacquet–Langlands perspective, and provides finiteness and growth results for Hecke eigensystems. These results create a coherent arithmetic picture linking moduli spaces, modular forms, supersingular phenomena, and inner-form correspondences, with potential implications for congruences and Galois representations in function field arithmetic.

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 which are also Hecke eigenforms. Among them we obtain generalized Hasse invariants which 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 $GL_r$.

Arithmetic Satake compactifications and algebraic Drinfeld modular forms

TL;DR

This work develops an arithmetic framework for Drinfeld modular varieties by constructing the arithmetic Satake compactification of rank Drinfeld moduli over and showing the universal Drinfeld module extends to a generalized Drinfeld module on the compactification. It defines algebraic Drinfeld modular forms over general bases through global sections of the ample line bundle , and analyzes their Hecke actions, including eigenforms modulo . The paper introduces generalized Hasse invariants to stratify the special fiber, connects supersingular Drinfeld modules to algebraic modular forms on inner forms of via a mod Jacquet–Langlands perspective, and provides finiteness and growth results for Hecke eigensystems. These results create a coherent arithmetic picture linking moduli spaces, modular forms, supersingular phenomena, and inner-form correspondences, with potential implications for congruences and Galois representations in function field arithmetic.

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 which are also Hecke eigenforms. Among them we obtain generalized Hasse invariants which 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 ) between Hecke eigensystems of rank Drinfeld modular forms and those of algebraic modular forms (in the sense of Gross) attached to a compact inner form of .

Paper Structure

This paper contains 15 sections, 43 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Drinfeld modules and moduli spaces
  3. The arithmetic Satake compactification
  4. Drinfeld modular forms and Hecke operators
  5. Drinfeld modular forms over $A[{\mathfrak{n}}^{-1}]$
  6. Drinfeld modular forms over $A_{(v)}$
  7. Systems of Hecke eigenvalues in $\overline \mathbb F_v$
  8. The Supersingular locus and Hecke modules
  9. The $v$-rank stratification
  10. Supersingular Drinfeld modules
  11. Supersingular and algebraic Drinfeld modular forms mod $v$
  12. Generalized Hasse invariants and $v$-rank strata
  13. Coefficient modular forms
  14. Generalized Hasse invariants
  15. Hecke eigensystems of Drinfeld modular forms modulo $v$

Key Result

Theorem 1.1

For every fine open compact subgroup $K=K_v K^v\subset G(\mathbb A^\infty)$, where $K_v=G(A_v)$ and $K^v\subset G(\mathbb A^{v\infty})$, the Drinfeld moduli scheme ${\bf M}^r_K$ over $A_{(v)}$ of rank $r$ and level $K$ possesses an arithmetic Satake compactification $\overline{{\bf M}}^r_K$ projecti

Theorems & Definitions (109)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: pink
  • Definition 2.2: pink
  • Theorem 2.3
  • Definition 2.4
  • Proposition 2.5
  • ...and 99 more