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Arithmetic Quotients of the Bruhat-Tits Building for Projective General Linear Group in Positive Characteristic
About this Title
Satoshi Kondo and Seidai Yasuda
Publication: Memoirs of the American Mathematical Society
Publication Year:
2025; Volume 306, Number 1547
ISBNs: 978-1-4704-7313-6 (print); 978-1-4704-8055-4 (online)
DOI: https://doi.org/10.1090/memo/1547
Published electronically: January 27, 2025
Keywords: Modular symbol,
automorphic representation,
positive characteristic,
Bruhat-Tits building
Table of Contents
Chapters
- 1. Introduction
- 2. Generalized simplicial complexes and their (co)homology
- 3. The Bruhat-Tits building and apartments
- 4. Arithmetic subgroups and modular symbols
- 5. Automorphic Forms with Steinberg at infinity
- 6. Double cosets for automorphic forms
- 7. Proof of Theorem
- 8. Universal Modular Symbols
- 9. On finite $p$-subgroups of arithmetic subgroups
- 10. Some spectral sequences
- 11. Proof for universal modular symbols
- 12. Comparison of modular symbols
Abstract
Let $d \ge 1$. We study a subspace of the space of automorphic forms of $\mathrm {GL}_d$ over a global field of positive characteristic (or, a function field of a curve over a finite field). We fix a place $\infty$ of $F$, and we consider the subspace $\mathcal {A}_{\mathrm {St}}$ consisting of automorphic forms such that the local component at $\infty$ of the associated automorphic representation is the Steinberg representation (to be made precise in the text).
We have two results.
One theorem (Theorem 5.4.2) describes the constituents of $\mathcal {A}_{\mathrm {St}}$ as automorphic representation and gives a multiplicity one type statement.
For the other theorem (Theorem 4.5.1), we construct, using the geometry of the Bruhat-Tits building, an analogue of modular symbols in $\mathcal {A}_{\mathrm {St}}$ integrally (that is, in the space of $\mathbb {Z}$-valued automorphic forms). We show that the quotient is finite when a level is fixed and give a bound on the exponent of this quotient.
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