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Eigenforms and graphs of Hecke operators with wild ramification

Rudrendra Kashyap, Vladyslav Zveryk

Abstract

Hecke operators on moduli of bundles over a global function field become substantially more complicated in the presence of ramification. We show that far enough in the Harder-Narasimhan cone of $\mathrm{Bun}_G$, this extra complexity has a simple structure, which allows to reduce the main study to the unramified case. Using the theory of graphs of Hecke operators, we transform this statement into a combinatorial condition. Utilizing the combinatorial language, we obtain tight bounds, and for generic eigenvalues exact formulas for the dimensions of Hecke eigenspaces with arbitrary ramification for $\mathrm{Bun}_{\mathrm{PGL}_2}$. Moreover, our methods allow to construct eigenforms explicitly.

Eigenforms and graphs of Hecke operators with wild ramification

Abstract

Hecke operators on moduli of bundles over a global function field become substantially more complicated in the presence of ramification. We show that far enough in the Harder-Narasimhan cone of , this extra complexity has a simple structure, which allows to reduce the main study to the unramified case. Using the theory of graphs of Hecke operators, we transform this statement into a combinatorial condition. Utilizing the combinatorial language, we obtain tight bounds, and for generic eigenvalues exact formulas for the dimensions of Hecke eigenspaces with arbitrary ramification for . Moreover, our methods allow to construct eigenforms explicitly.
Paper Structure (14 sections, 27 theorems, 188 equations, 6 figures)

This paper contains 14 sections, 27 theorems, 188 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Graph of Hecke operators and notation
  3. Transition between adelic and geometric sides
  4. Hecke correspondences for $\mathrm{GL}_n$
  5. Hecke correspondences for general $G$
  6. The $\mathrm{GL}_n$ case
  7. Geometry of Hecke graphs in the cusp of $\mathrm{Bun}_G$
  8. Generalities on infinite graphs
  9. Spaces of eigenforms for $\mathrm{PGL}_2$ unramified at $x$
  10. Spaces of eigenforms for $\mathrm{PGL}_2$ ramified at $x$
  11. Spaces of eigenforms for $\mathrm{PGL}_n$
  12. General Reductive Case
  13. Notations and general assumptions
  14. Generalizations of the $\mathrm{GL}_n$ results

Key Result

Proposition 3.2

Choose $(\mathcal{E},a)\in\mathrm{Bun}_{G, D}(\mathbb{F}_q)$. Edges $(\mathcal{E},a)\to(\mathcal{E}',b)$ in $\mathrm{Bun}_{G, D}(\mathbb{F}_q)$ for the Hecke operator $\Phi_x^{\omega_r}$ are in bijection with equivalence classes of exact sequences such that

Figures (6)

  • Figure 1: Graph for $\mathrm{PGL}_2$ unramified at $x$
  • Figure 2: $\infty$-part of the $\mathrm{PGL}_2$-graph ramified at $d[x]$
  • Figure 3: $\mathrm{PGL}_2$-graph ramified at $d[x]$, where $|k_x|=q^r$
  • Figure 4: Computation of eigenspaces for $\mathrm{PGL}_2$ ramified at $x$
  • Figure 5: Ramified graph for $G=\mathrm{PGL}_2$ and $X=\mathbb{P}^1$ at $1\cdot[x]$, $\deg x=1$
  • ...and 1 more figures

Theorems & Definitions (66)

  • Remark 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • Definition 4.1
  • ...and 56 more