What is the origin of the term "modular" in different areas?

Question

The word "modular" is used in many different, seemingly unrelated contexts. For example, modular forms, modular representation theory, modular latices, moduli spaces, modules, and modular arithmetic. Why is the word modular (and it's derivatives) so ubiquitous in mathematics, in seemingly unrelated areas?

Answer 1

Modular in mathematics means "relating to a modulus", which is essentially some form of "reduction of information" or a "projection" from a richer set of numbers to a less rich one, by virtue of an equivalence relation.

Positive and negative numbers projected down to positive numbers only.

Modular arithmetic: the integers, projected down to a set of non-negative integers less than N.

etc.

Answer 2

The sense of "modular" in (elliptic, Hilbert-Blumenthal, Siegel, hermitian, ...) "modular forms", and in "moduli space" appears to be connected to a somewhat archaic use of "modul" (in German, mid-to-late 19th century) for what would be "lattice" (in a Euclidean space) in English, like $\mathbb Z\oplus \mathbb Z\subset \mathbb R^2$. This arose c. 1800 first in the study of elliptic curves over $\mathbb C$, and then "abelian" functions by Abel and Jacobi. The space of "modules", that is, the parameter space for elliptic curves, was the "modul-space" (in some linguistic form). This was a good part of Siegel's studies in the late 1930s regarding moduli of (e.g.) principally polarized abelian varieties (over $\mathbb C$). Functions and sections of line bundles on the moduli-space are "modular..."

Rings of integers in algebraic number fields were often modeled in more "geometric" fashion as lattices ("modules") sitting in Euclidean spaces, and this transitional viewpoint is still used in introductory treatments nowadays. This also connects to "geometry of numbers" developed by Minkowski and others c. 1900.

I think unrelated to the previous: on a topological group, "the modular function" tells how right (or left) Haar measure changes under left (or right) translation. This might be more etymologically related to the use as in "modular arithmetic", but I do not know. I cannot easily determine the early use of "modular" in reference to Gauss' invention of "congruences".