Where the Slopes Are

For each prime number p which does not divide N there is a linear operator Tp acting on Sk(N,C), known as the p-th Hecke operator. (In fact, the Tp stabilize Sk(N,Q).) A modular form which is an eigenvector for all of these linear operators simultaneously is called an eigenform; the space Sk(N,C) has a basis made up of eigenforms, and the Fourier coefficients of these eigenforms can be normalized (by requiring a1 = 1) to belong to a finite extension of Q. The eigenvalues of the Tp operator encode significant arithmetic information about the modular form and various other objects which can be attached to it (for example, a Galois representation). In our setting, the eigenvalue of Tp acting on an eigenform f ∈ Sk(N,C) is a totally real algebraic number whose absolute value (with respect to any embedding of Q into C) is between −2p(k−1)/2 and 2p(k−1)/2. If we normalize the eigenvalues by dividing by p(k−1)/2, the normalized eigenvalues are real numbers in the interval [−2, 2], and we can ask about their distribution in that interval. The Sato-Tate Conjecture, still very