The zeroes of period polynomials -

Abstract

A modular form is a function meromorphic in the upper half plane and at the cusps. It satisfies certain transformation conditions under the full modular group The space of entire modular forms of integer weight on the full group is finite dimensional and thus the Fourier coefficients of these forms possess interesting arithmetical properties. Moreover, the zeroes of modular forms have been studied intensively and were the center of attention in the field. As an example, we show that the zeroes of the Eisenstein series of weight greater or equal to 4 lie on the portion of the unit circle. The (k+1) fold integral of a modular form gives rise to what is known as the period polynomials. These polynomials satisfy certain consistency condition and have interesting connections to L-functions. We show that the zeroes of period polynomials lie on the unit circle.