Application: Quadratic Forms

In this chapter, we study the generating functions of representations of quadratic forms, which are well known to be modular forms. In Sects. 3.1 and 3.2, we study the problem in its most general form. In Sect. 3.3, we work on a family of binary quadratic forms. Much of the previous work in representations of quadratic forms concerns diagonal quadratic forms. In Sect. 3.4, we specialize the quadratic form to be a diagonal form and give a result that generalizes and unifies almost all the previous results concerning formulas for the number of representations of diagonal quadratic forms, see Theorem 3.4.3. We use our results to give a SAGE function that computes the Eisenstein part of any diagonal quadratic form with an even number of variables. Section 3.4.4 is devoted to explicit examples.