Borcherds Forms and Generalizations of Singular Moduli

Abstract

In the first part of this thesis, we prove an explicit formula for the average of a Borcherds form over CM points associated to a quadratic form of signature (n, 2). One step in the proof extends a theorem of Kudla to the case n = 0. The formula we obtain involves the negative Fourier coefficients of a modular form F, and the second terms in the Laurent expansions (at s = 0) of the Fourier coefficients of an Eisenstein series of weight one. These Laurent expansion terms were calculated by Kudla, Rapoport and Yang in a special case. We extend their results to a more general case.

In the second part of this thesis, we consider examples of our main theorem for n = 0 and n = 1 in more detail. When n = 0, we let k be an imaginary quadratic field and we obtain a function on the product of the ideal class group of k with the squares of the ideal class group of k. The example for n = 1 allows us to reproduce the well-known singular moduli result of Gross and Zagier. This result gives an explicit factorization of the function J(D, d), defined as a product of j(z)-j(w) over points z and w of discriminant D and d, respectively, where D and d are negative relatively prime fundamental discriminants.