For odd prime numbers p , I present a conjectural formula for the T p -slopes of classical modular forms of level coprime to p , based on empirical observations. To provide a structural underpinning for the conjecture, I first develop some results about modular forms mod p . In particular, a certain semisimplification of the Hecke module of modular forms mod p decomposes into a direct sum of finitely many cyclic submodules generated in weights ≤2 p + 1 over a noncommutative extension of the Hecke algebra. The forms in each of these submodules all have the same associated Galois representation up to twisting by the cyclotomic character. I attach abstract slope sequences to the forms in these submodules and, provided that the associated Galois representations are locally reducible at p , I conjecture that these abstract slopes agree with the actual slopes associated to classical modular forms with the same associated residual Galois representations.
My Research