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arXiv · 2603.15931

Eigenforms and graphs of Hecke operators with wild ramification

Rudrendra Kashyap·Vladyslav Zveryk
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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 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 $\mathrm{Bun}_{\mathrm{PGL}_2}$. We compare these formulas to the known results in the theory of Eisenstein series. Moreover, our methods allow to construct eigenforms explicitly.

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Rudrendra Kashyap, Vladyslav Zveryk. 2026-03-16. Eigenforms and graphs of Hecke operators with wild ramification. https://arxiv.org/abs/2603.15931

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