Lower bounds on dimensions of mod-p Hecke algebras: The nilpotence method

We present a new method for obtaining lower bounds on the Krull dimension of a local component of a Hecke algebra acting on modular forms modulo a prime p of level one and all weights at once. This so-called nilpotence method proceeds by showing that the Hilbert-Samuel function of the Hecke algebra, which is a noetherian local ring, grows fast enough to establish a lower bound on dimension. By duality it suffices to exhibit enough forms annihilated by a power of the maximal ideal. We use linear recurrences associated with Hecke operators to reduce the problem of finding these many annihilated forms to a purely algebraic question about the growth of nilpotence indices of recurrence operators on polynomial algebras in characteristic p. Along the way we introduce a theory of recursion operators over any field. The key technical result is the Nilpotence Growth Theorem for locally nilpotent recursion operators over a finite field; its proof is elementary and combinatorial in nature.;The nilpotence method currently works only for the small primes p for which the modular curve X0( p) has genus zero (p = 2, 3, 5, 7, 13) but we sketch a plan for generalizing it to all primes and all levels.