An Introduction to Classical Modular Forms with Kohnen’s Proof of the Product Expansion of the Delta Function
This pro ject studies modular forms, a certain family of complex analytic functions which have an invariance property making them very useful in analytic number theory. The particular focus of this paper is on a speciﬁc modular form, namely the delta function ∆(z ). This function is usually deﬁned as an inﬁnite sum, but it is equally well expressed as a certain inﬁnite product. Several challenging and high-level proofs of the equivalence of these two deﬁnitions have existed for most of the last century, but a new and surprising proof due to Kohnen has recently been published in , which uses an interesting new technique. This purpose of this pro ject has been to understand this proof. To do this, we will develop the basic theory of modular forms, including results on linear fractional transformations and the modular group, leading up to the deﬁnition of a modular form. We proceed with some examples of modular forms of low integer weights, namely the Eisenstein series and the delta function. We also discuss the powerful Hecke operators; Kohnen’s proof of the product expansion of the delta function rests on a multiplicative analogue of the (usually additive) Hecke operators. The main proof will then be presented, in a more explicit fashion than in the very terse note by Kohnen. It is unclear whether the same method can be used eﬀectively on other modular forms, but we discuss these possibilities in a conclusion.
Franklin and Marshall College Archives, Undergraduate Honors Thesis 2010
- F&M Theses Collection