Theta Series and its Application to Sums of Squares

Abstract

Let Q be a positive definite quadratic form on Z^k. Consider the Theta Function defined by ∑e(Q(m) z) for some z in the upper half plane. As an interesting application of Modular Forms, we study the number of representations of an integer s by Q. In this regard, we begin by proving the transformation law of Theta following Goro Shimura's approach of the proof which uses some essential techniques such as the Poisson Summation Formula and Fourier Transforms. This shows that Theta is a modular form of weight k/2 on some congruence subgroup. After that, we study the Eisenstein series of weight k ≥ 3 on Γ(M), as well as write its Fourier expansion used in expressing bases of the spaces of modular forms accordingly. To end, we approach the growth of Theta's Fourier coefficients to obtain asymptotic formulas for the number of representations mentioned above.