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 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 . Moreover, our methods allow to construct eigenforms explicitly.
Keywords
Cite
@article{arxiv.2603.15931,
title = {Eigenforms and graphs of Hecke operators with wild ramification},
author = {Rudrendra Kashyap and Vladyslav Zveryk},
journal= {arXiv preprint arXiv:2603.15931},
year = {2026}
}
Comments
45 pages, 8 figures. 1. Fixed Theorems 4.16 and 5.15 by adding one point to ramification at x. Showed a nontrivial torus-monodromy appearing otherwise (Theorems 4.17, 5.16, Example 4.32). Added a graph covering formalism (Definition 4.14, Lemmas 4.15, 5.14). Extended main results to forgetting D_1 in D_1+D_2. 2. Improved Example 4.31, corrected PGL_2-formulas. 3. Rewrote Introduction