On Shifted Convolution Sums Involving the Fourier Coefficients of Theta Functions Attached to Quadratic Forms

The study of shifted convolution sums has acquired a prominent place in current number theory research owing to its potential applications to the sub-convexity problem, while quadratic forms have fascinated mathematicians since antiquity. This thesis, which deals with both these topics, studies shifted convolution sums involving the Fourier coefficients of Theta Series associated to a positive definite integral quadratic form and a cuspidal Hecke eigenform of integral weight. Our aim is to generalize the work of W. Luo, J. Hafner, and H. Iwaniec et al. in this new setting. Three independent approaches are used in this endeavour -- the spectral theory of the hyperbolic Laplacian, the delta-symbol method (a variant of the Hardy-Littlewood-Ramanujan circle method), and the theory of Poincare series via a Poisson-Voronoi summation formula. We establish asymptotic formulae in all three aspects with the spectral theory approach providing the optimal estimate for the error term when one of the forms involved is cuspidal, while the delta-symbol method gives a sharp error term when only Theta Series are involved in the shifted convolution sum.