Riemann hypothesis for period polynomials of modular forms

摘要

The period polynomial r(f) (z) for an even weight k >= 4 newform f epsilon S-k(Gamma(0)(N)) is the generating function for the critical values of L(f, s). It has a functional equation relating r(f) (z) to r(f) (-1/Nz). We prove the Riemann hypothesis for these polynomials: that the zeros of r(f) (z) lie on the circle vertical bar z vertical bar = 1 / root N . We prove that these zeros are equidistributed when either k or N is large.