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.

Dec 3, 2025 · 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).

odular transfor-mations arise naturally. A lot of the time, modular symmetry doesn't have to be exactly satis ed in t e physics applications, but only nearly. For instance, if you have a function …

In order to motivate the definition of the Hecke operators on modular forms, we first define them in terms of lattices, following the presentation in [13, VII.5.1].

4 days ago · Modular functions have enabled the calculation of billions of digits of $\pi$. The proof of Fermat's Last Theorem rests on the connection between modular and elliptic functions. …

Feb 7, 2011 · The study of modular functions began in the 19th century in connection with the study of elliptic functions and preceded the appearance of the general theory of automorphic …

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 = …

Jun 13, 2025 · Modular functions are a fundamental concept in number theory, playing a crucial role in various areas of mathematics and computer science. In this article, we will delve into …

It is easier to construct functions that transform in a certain way under the action of Γ(N); the quotient of two such functions of same type will then be a modular function.