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Today Klein's $j$-invariant is used in various context's, the most famous one being maybe "Monstrous Moonshine". But what was the original motivation for the study of the $j$-invariant?

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  • $\begingroup$ I would guess it first arose in the theory of elliptic curves, where it distinguishes between nonisomorphic elliptic curves (over C, for historical purposes of first appearance). Back in the 19th century, elliptic curves, elliptic functions, and modular functions were all mixed together. $\endgroup$ – KCd Jul 26 '16 at 14:59
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This is an example of absolute invariant in the context of classical invariant theory. Of course it is of paramount importance in the theory of elliptic curves but it is not just about them. Such a curve can be written as $y^2=f(x)$ where $f$ is a nonhomogenous polynomial of degree four (or three). By homogenization this corresponds to a binary quartic $$ F(x_1,x_2)=x_2^4 f\left(\frac{x_1}{x_2}\right)= a_0x_1^4+4 a_1 x_1^3 x_2+ 6 a_2 x_1^2 x_2^2+4 a_3 x_1 x_2^3+a_4 x_2^4\ . $$ The ring of $SL_2$ invariants is generated by $$ S= a_0a_4 - 4a_1 a_3 + 3a_2^2 $$ and $$ T=a_0a_2a_4 - a_0a_3^2 - a_1^2a_4 + 2a_1a_2a_3 - a_2^3 $$ while the $j$ invariant is given by $$ j=\frac{S^3}{S^3-27T^2}\ . $$ It gives a way of parametrizing $SL_2$ orbits of such binary quartics (see this MO post for details) and thus of isomorphism classes of elliptic curves. Early work on this is in papers by Eisenstein, Boole and Cayley from the first half of the 19th century.

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