On the semi-simplicity of the $U_p$-operator on modular forms

Abstract

Let $p$ be a prime number and $N$ an integer prime to $p$. We show that the operator $U_p$ on the space of cuspidal modular forms of level $pN$ and weight two is semi-simple. It follows from this that the Hecke algebra acting on the space of weight two forms of level $M$ is reduced if $M$ is cube free. Assuming Tate's conjecture for cycles on smooth projective varieties over finite fields, we generalize these results to higher weights. The main point in the proof is that the crystalline Frobenius of the reduction mod $p$ of the motive associated to a newform of level prime to $p$ and weight at least two cannot be a scalar. Assuming Tate's conjecture, it follows that Ramanujan's inequality is strict. For $N$ prime, we relate the discriminant of the weight two Hecke algebra to the height of the modular curve $X_0(N)$, for which we get an upper bound.