Modulgrang

Modulgrang is a term used in the field of mathematics, specifically in the study of modular forms and modular curves. It refers to the quotient of a modular curve by a congruence subgroup of the modular group. Modular curves are Riemann surfaces that parameterize certain types of elliptic curves, and they are closely related to modular forms, which are complex-analytic functions on the upper half-plane with certain transformation properties under the action of the modular group.

The modular group, denoted by SL(2, Z), is the group of 2x2 matrices with integer entries and determinant 1. A congruence subgroup is a subgroup of the modular group that is defined by congruence conditions on the matrix entries. For example, the principal congruence subgroup of level N, denoted by Gamma(N), consists of matrices in SL(2, Z) that are congruent to the identity matrix modulo N.

The modulgrang of a modular curve X by a congruence subgroup Gamma is the quotient space X/Gamma, which is again a Riemann surface. The study of modulgrangs is important in number theory, as they are related to the arithmetic properties of elliptic curves and modular forms. For example, the j-invariant of an elliptic curve can be viewed as a function on the modulgrang of the modular curve X(1) by the full modular group SL(2, Z).

In summary, modulgrang is a mathematical concept that arises in the study of modular forms and modular curves. It refers to the quotient of a modular curve by a congruence subgroup of the modular group, and it plays an important role in number theory and the study of elliptic curves.