Eigenforms and graphs of Hecke operators with wild ramification

2026/03/31 by Rudrendra Kashyap, Vladyslav Zveryk · 1 citation
Mathematics · #math.AG #math.NT #math.RT

paper · pdf

48 pages, 8 figures. Added connection to Eisenstein series (Section 4.7) and explicit bases of generalized eigenspaces in terms of Eisenstein series (Section 4.5)

arxiv created 2026/07/30 · arxiv updated 2026/07/31

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 Bun_G, this extra complexity has a simple structure, which allows to reduce most of the 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 Bun_PGL2. We compare these formulas to the known results in the theory of Eisenstein series. Moreover, our methods allow to construct eigenforms explicitly.

Citations

• Graphs of Hecke operators in mixed ramification
• Hecke modifications of vector bundles
• Hecke eigenspaces for the projective line
• An explicit geometric Langlands correspondence for the projective line minus four points
• Hall algebras and graphs of Hecke operators for elliptic curves
• Epipelagic representations and rigid local systems
• The Harder-Narasimhan stratification of the moduli stack of G-bundles via Drinfeld's compactifications
• Graphs of Hecke operators
• Kloosterman sheaves for reductive groups
• Toroidal automorphic forms for some function fields
• Geometric Langlands duality and representations of algebraic groups over commutative rings
• Coherent sheaves with parabolic structure and construction of Hecke eigensheaves for some ramified local systems

Cited by

• Graphs of Hecke operators in mixed ramification

Related

• Graphs of Hecke operators in mixed ramification
• Hecke eigenspaces for the projective line
• Graphs of Hecke operators
• Rational eigenfunctions of the Hecke operators
• Topological Hecke eigenforms