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Schwartz $κ$-densities on the moduli stack of rank $2$ bundles near stable bundles

David Kazhdan, Alexander Polishchuk

TL;DR

The work addresses the boundedness of Schwartz $| ext{ω}|^{kap}$-densities on the moduli stack of rank-$2$ bundles with odd determinant over a non-archimedean curve, connecting analytic questions to algebro-geometric invariants. The authors formulate Conjectures A, A', B, and C and reduce boundedness questions to rational-singularity and flatness properties of Brill–Noether-type loci via the Bertram–Thaddeus construction and push-forwards of densities, with the Aizenbud–Avni theorem providing the analytic bridge. They verify the framework in genus $2$ and in non-hyperelliptic genus $3$, proving boundedness/continuity results for genus-$2$ and partial but substantial results for genus-$3$ through explicit degeneracy-locus analyses, rational singularities, and resolution chains. The results advance the analytic Langlands program by showing that Hecke-related densities on ${ m Bun}_{Lambda_0}$ have well-behaved push-forwards to ${ m f M}_{Lambda_0}$ in key low-genus cases, tying geometric properties to distributional continuity and $L^2$-type interpretations. The methods combine a careful atlas of algebraic geometry (degeneracy loci, birational transformations, and resolution theory) with harmonic-analysis notions (Schwartz spaces, bounded distributions) to create a robust framework for extending density-theoretic constructions across moduli stacks.

Abstract

Let $C$ be a curve over a non-archimedean local field of characteristic zero. We formulate algebro-geometric statements that imply boundedness of functions on the moduli space of stable bundles of rank $2$ and fixed odd degree determinant over $C$, coming from the Schwartz space of $κ$-densities on the corresponding stack of bundles (earlier we proved that these functions are locally constant on the locus of very stable bundles). We prove the relevant algebro-geometric statements for curves of genus $2$ and for non-hyperelliptic curves of genus $3$.

Schwartz $κ$-densities on the moduli stack of rank $2$ bundles near stable bundles

TL;DR

The work addresses the boundedness of Schwartz -densities on the moduli stack of rank- bundles with odd determinant over a non-archimedean curve, connecting analytic questions to algebro-geometric invariants. The authors formulate Conjectures A, A', B, and C and reduce boundedness questions to rational-singularity and flatness properties of Brill–Noether-type loci via the Bertram–Thaddeus construction and push-forwards of densities, with the Aizenbud–Avni theorem providing the analytic bridge. They verify the framework in genus and in non-hyperelliptic genus , proving boundedness/continuity results for genus- and partial but substantial results for genus- through explicit degeneracy-locus analyses, rational singularities, and resolution chains. The results advance the analytic Langlands program by showing that Hecke-related densities on have well-behaved push-forwards to in key low-genus cases, tying geometric properties to distributional continuity and -type interpretations. The methods combine a careful atlas of algebraic geometry (degeneracy loci, birational transformations, and resolution theory) with harmonic-analysis notions (Schwartz spaces, bounded distributions) to create a robust framework for extending density-theoretic constructions across moduli stacks.

Abstract

Let be a curve over a non-archimedean local field of characteristic zero. We formulate algebro-geometric statements that imply boundedness of functions on the moduli space of stable bundles of rank and fixed odd degree determinant over , coming from the Schwartz space of -densities on the corresponding stack of bundles (earlier we proved that these functions are locally constant on the locus of very stable bundles). We prove the relevant algebro-geometric statements for curves of genus and for non-hyperelliptic curves of genus .
Paper Structure (17 sections, 42 theorems, 159 equations)

This paper contains 17 sections, 42 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Push-forwards of Schwartz $\kappa$-densities
  3. Background on Schwartz spaces and distributions
  4. Push-forwards for varieties
  5. Push-forwards for admissible stacks and an extension result
  6. From algebraic geometry to results on Schwartz $\kappa$-densities
  7. Bertram-Thaddeus construction
  8. The relation between line bundles
  9. The proof of Theorem \ref{['main-conj-thm']}
  10. Towards Conjecture B
  11. Degeneracy loci and their partial resolutions
  12. Irreducibility of $F_E$ and Conjecture C
  13. On some cases of Conjecture C
  14. Singularities of $F_E$: easy cases
  15. Proof of Conjecture B for genus $2$
  16. ...and 2 more sections

Key Result

Lemma 2.3

(i) If $\nu\in {\mathcal{D}}(X(k),|L|^{\kappa})$ is bounded then for any $g\in C^\infty(X(k))$, $\nu\cdot f$ is also bounded. (ii) Let $X$ be a smooth $k$-variety, $U\subset X$ an open subvariety, $\nu\in {\mathcal{D}}(X(k),|L|^{\kappa})$ a bounded element, such that $\nu|_U$ is locally constant. Th

Theorems & Definitions (91)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 81 more