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 22, 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).

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 the …

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 …

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 …

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.

Modular Forms and L-functions Lectures 1. Characters Character: Let G be abelian topological group. A (unitary) Character of G is a continuous homomorphism: : G ! U(1) = fz 2 C : jzj = 1g Notation: The …

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 …

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.