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 de ned on a …

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.

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 functions.

Let f be a weakly modular function of weight k and level Γ(1). Then there exists a unique meromorphic function ̃f in D∗(0, 1) such that f = ̃f e2πiτ.

modular transformations λ: 1. −1/τ : L0 is mapped onto a circle C0 tangent to the point τ. = 0 in the upper half-plane. Clearly, the “smaller” the circle is when the. larger the t0 > 0 is chosen; 2. 1−1/τ : L0 is …

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.

Feb 9, 2018 · The function Δ is called the modular function of G (notice that, by uniqueness up to scalar multiple of left Haar measures, Δ only depends on G). Other names for Δ that can be found are: Haar …