A modular function is a function that is invariant with respect to the modular group, but without the condition that it be holomorphic in the upper half-plane (among other requirements).

It is easy to define modular functions and forms, but less easy to say why they are important, especially to number theorists. Thus I shall begin with a rather long overview of the subject.

w notable applications of modular forms. These are only for illustrative purposes, so if you don't understand the details of these (and you likely won't until we are much further in the class

6 days ago · Every rational function of Klein's absolute invariant J is a modular function, and every modular function can be expressed as a rational function of J (Apostol 1997, p. 40).

Every elliptic curve E=Q is modular. In this lecture we will explain what it means for an elliptic curve over Q to be modular (we will also define the term semistable). This requires us to delve briefly into …

Since e2πiτ is locally biholomorphic, we see that ̃f is meromorphic. Uniqueness follows from the surjectivity of e2πiτ. If f is a weakly modular function of weight k and level Γ(1), we say that f is …

λ cz + d = λ(z), where the Möbius transformation belongs to the modular group is called an automorphic function. Recall that for a given Weierstrass elliptic function ℘(z), we have e1 = ℘(ω1/2), e2 = …

6 days ago · Modular forms satisfy rather spectacular and special properties resulting from their surprising array of internal symmetries. Hecke discovered an amazing connection between each …

It is for this reason that the j-function is sometimes referred to as the modular function. Indeed, every modular function for (1) = SL 2(Z) can be expressed in terms of the j-function.

May 27, 2025 · In this article, we will explore the definition and properties of modular functions, their connections to modular forms, and their applications in number theory and algebraic geometry.