Department of Mathematics - Master Calendar

Speaker: Jeremy Rouse (Wake Forest University)

Title: Modular forms with only nonnegative coefficients

Abstract: Many modular forms are generating functions for interesting arithmetic objects. For this reason, it is natural to ask, given a modular form f, whether all the Fourier coefficients of f are non-negative. We study this problem for modular forms of level 1 and weight k ≡ 0 (mod 4). We define A(k) to be the smallest positive integer so that if f(z) = Σ a(n) q^n is any weight k modular form for which a(r) ≥ 0 for all r ≤ A(k), then all the Fourier coefficients of f are non-negative. We show that k^2 << A(k) << k^4(log k)^2.

We also show that the set of modular forms with non-negative Fourier coefficients corresponds naturally to the points in a finite, bounded polytope, and discuss challenges in considering analogous questions in higher level. This is joint work with Paul Jenkins.