Number Theory Seminar

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A basic goal of the theory of automorphic forms is to estimate sums of Hecke eigenvalues. In this talk, we consider a finer version of this problem, where the sum is taken over the values of an integer polynomial. We give new upper bounds for these sums under certain natural assumptions. Our results are unconditional when the degree of the polynomial in question is at most four. We also examine the analogous sum over polynomials in several variables. In particular, we obtain an estimate for all cubic polynomials in two variables that define elliptic curves. The talk is based on joint work with Liyang Yang.