[Paper Review] Graphs of Hecke operators

Oliver Lorscheid|arXiv (Cornell University)|Dec 16, 2010

Advanced Algebra and Geometry5 references3 citations

TL;DR

This paper introduces a novel graph-theoretic framework for Hecke operators on function fields, modeling their action via graphs that unify unramified automorphic forms and reduction theory. It establishes that these graphs decompose into a finite nucleus and finitely many cusps, enabling a new proof of finite-dimensionality for unramified cusp forms and toroidal automorphic forms through linear conditions on graph functions.

ABSTRACT

Let $X$ be a curve over $\F_q$ with function field $F$. In this paper, we define a graph for each Hecke operator with fixed ramification. A priori, these graphs can be seen as a convenient language to organize formulas for the action of Hecke operators on automorphic forms. However, they will prove to be a powerful tool for explicit calculations and proofs of finite dimensionality results. We develop a structure theory for certain graphs $G_x$ of unramified Hecke operators, which is of a similar vein to Serre's theory of quotients of Bruhat Tits trees. To be precise, $G_x$ is locally a quotient of a Bruhat Tits tree and has finitely many components. An interpretation of $G_x$ in terms of rank 2 bundles on $X$ and methods from reduction theory show that $G_x$ is the union of finitely many cusps, which are infinite subgraphs of a simple nature, and a nucleus, which is a finite subgraph that depends heavily on the arithmetics of $F$. We describe how one recovers unramified automorphic forms as functions on the graphs $G_x$. In the exemplary cases of the cuspidal and the toroidal condition, we show how a linear condition on functions on $G_x$ leads to a finite dimensionality result. In particular, we re-obtain the finite-dimensionality of the space of unramified cusp forms and the space of unramified toroidal automorphic forms. In an Appendix, we calculate a variety of examples of graphs over rational function fields.

Motivation & Objective

• To develop a global geometric framework for Hecke operators on function fields, overcoming limitations of strong approximation in cases with even class number or even-degree places.
• To provide a combinatorial model—graphs of Hecke operators—for the action of unramified Hecke operators on automorphic forms.
• To establish a structure theory for these graphs analogous to Serre’s theory of quotients of Bruhat-Tits trees.
• To prove finite-dimensionality of spaces of unramified cusp forms and toroidal automorphic forms using linear conditions on graph functions.
• To connect the graph structure to rank-2 vector bundles and reduction theory over the curve X.

Proposed method

• Define a graph $\mathcal{G}_x$ for each unramified Hecke operator at a place $x$, with vertices corresponding to double cosets $\Gamma \backslash \mathrm{PGL}_2(F_x)/\mathrm{PGL}_2(\mathcal{O}_x)$.
• Construct edges based on adjacency in the Bruhat-Tits tree $\mathcal{T}_x$, using stabilizers and coset representatives.
• Use reduction theory for rank-2 vector bundles on $X$ to interpret the graph structure geometrically.
• Label vertices via a vertex labelling map to relate graph components to ideal classes and arithmetic of $F$.
• Decompose $\mathcal{G}_x$ into a finite nucleus (arithmetic-dependent) and finitely many infinite cusps (geometrically simple).
• Model automorphic forms as functions on $\mathcal{G}_x$, with Hecke operators acting via adjacency sums.

Experimental results

Research questions

• RQ1 How can Hecke operators on automorphic forms over global function fields be represented geometrically beyond the local $p$-adic framework?
• RQ2 What is the global structure of the graph associated to an unramified Hecke operator, and how does it relate to the arithmetic of the function field?
• RQ3 Can the finite-dimensionality of spaces of unramified cusp forms be re-derived using this graph-theoretic model?
• RQ4 How do the components of the graph—nucleus and cusps—reflect the geometry and arithmetic of the underlying curve and function field?
• RQ5 To what extent can linear conditions on functions over the graph recover known automorphic form spaces?

Key findings

• The graph $\mathcal{G}_x$ of an unramified Hecke operator is locally a quotient of a Bruhat-Tits tree and has finitely many connected components.
• The graph $\mathcal{G}_x$ decomposes into a finite nucleus, which encodes the arithmetic of the function field $F$, and finitely many infinite cusps, each isomorphic to a ray.
• Unramified automorphic forms are realized as functions on $\mathcal{G}_x$ that satisfy a linear condition derived from the Hecke operator action.
• Finite-dimensionality of the space of unramified cusp forms is recovered by imposing a vanishing condition at the cusps, leading to a finite-dimensional solution space.
• Finite-dimensionality of the space of unramified toroidal automorphic forms is obtained via a linear condition on graph functions, analogous to the cusp condition.
• The theory applies universally to any global function field, even those with even class number or places of even degree, overcoming limitations of strong approximation.