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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?

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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.

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  • $\begingroup$ Per en.wikipedia.org/wiki/Modulus, from the latin modus meaning measure or manner. How would you extend your answer to physics/engineering contexts, such as "modulus of elasticity"? I agree that it can, could be a nice addendum though. $\endgroup$
    – erfink
    Commented Mar 17, 2017 at 19:16
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    $\begingroup$ I don't really see the connection for modular forms or modular representation theory. What is the modulus there? $\endgroup$
    – vukov
    Commented Mar 17, 2017 at 19:20
  • $\begingroup$ I would say that modulus of elasticity is the properties of something, stripped down to some essential measure, which is a subset of the full set of properties - again, a reduction in properties. Two things may have the same modulus of elasticity, though their breaking strains, and proneness to fatigue may be different. $\endgroup$ Commented Mar 17, 2017 at 19:20
  • $\begingroup$ @vukov i'm in danger of getting out of my depth but a modular function is is invariant with respect to the modular group and therefore there is an equivalence relation at play just like $-1$ can be considered equivalent to $+1$ under the appropriate equivalence relation, only with a much more complex equivalence $\endgroup$ Commented Mar 17, 2017 at 19:30
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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".

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