Nilpotence order growth of recursion operators in characteristic p

TL;DR

This paper investigates the growth rate of recursion operators in characteristic p and applies findings to bound the Krull dimension of certain Hecke algebras, extending previous results with new methods.

Contribution

It establishes that the killing rate of specific recursion operators grows slower than linearly, providing new bounds relevant to the structure of mod-p Hecke algebras.

Findings

Growth rate of recursion operators is sublinear in degree

Lower bounds for Krull dimension in genus-zero case

Application to p=2 and p=3 cases in level one

Abstract

We prove that the killing rate of certain degree-lowering "recursion operators" on a polynomial algebra over a finite field grows slower than linearly in the degree of the polynomial attacked. We also explain the motivating application: obtaining a lower bound for the Krull dimension of a local component of a big mod-p Hecke algebra in the genus-zero case. We sketch the application for p=2 and p=3 in level one. The case p=2 was first established in by Nicolas and Serre in 2012 using different methods.