Zeros of Modular Forms

Abstract

For p a prime larger than 7, the Eisenstein series of weight p-1 has some remarkable congruence properties modulo p, implying for example that the j-invariants of its zeros (which are known to be real algebraic numbers in the interval [0,1728]), are all modulo p at most quadratic over the field with p elements, are congruent modulo p to the zeros of certain truncated hypergeometric series. In my thesis, I introduce the “theta modular form" of weight k, defined as the unique modular form of that weight for which the first dim(M_k) Fourier coefficients are identical to those of the Jacobi theta series. Theta modular form modulo p relate to the average weight enumerators in coding theory. I show that theta modular forms of weight (p+1)/2 behave in many ways like Eisenstein series: the j-invariants of their zeros all belong to the interval [0,1728], are modulo p all in the ground field with p elements, and are congruent modulo p to the zeros of a truncated hypergeometric function (with parameters halved compared to the Eisenstein series).