Vol. 12, No. 3, 2018

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Nilpotence order growth of recursion operators in characteristic $p$

Anna Medvedovsky

Vol. 12 (2018), No. 3, 693–722
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.

Keywords
linear recurrences in characteristic $p$, modular forms modulo $p$, congruences between modular forms, $ \bmod p$ Hecke algebras, $p$-regular sequences, base representation of numbers
Mathematical Subject Classification 2010
Primary: 11T55
Secondary: 11B85, 11F03, 11F33
Milestones
Received: 27 July 2017
Revised: 12 January 2018
Accepted: 23 February 2018
Published: 12 June 2018
Authors
Anna Medvedovsky
Max-Planck-Institut für Mathematik
Bonn
Germany