June 11th, 2021

Some “gmachine1728” on the internet reminded me of the j-variant

Originally published at 狗和留美者不得入内. You can comment here or there.

https://en.gravatar.com/gmachine1728

1729 = 10^3+7^3 = 12^3+1^3 is a number associated with Ramanujan. 1728 is also uniquely mathematical, in that it appears in the j-variant of the theory of complex multiplication. Though I have no actual detailed knowledge on the aforementioned mathematical topic as of June 2021, I shall quote the following.

Strong convergence in artistic tastes amongst people at the extreme right tail. Another point of convergence is the theory of complex multiplication in math, involving the j-invariant, moduli spaces, and abelian extensions of imaginary quadratic number fields.

For example, the great mathematician Barry Mazur writing

The elliptic modular function is loved by the analysts, arithmeticians and algebraic geometers who study elliptic curves since the isomorphism class of the elliptic curve formed by the lattice generated by the complex numbers 1 and z is completely determined by j(z), usually referred to as the j-invariant of the elliptic curve. It is the showcase example of a modulus in algebraic geometry, i.e., a continuous parameter that classifies a continuously varying array of distinct isomorphism classes of mathematical objects.

AND it was loved by Leopold Kronecker who hitched the aspirations of his youth (his Jugendtraum) on the ability of the elliptic modular function to help generate, in a magically ordered way, all algebraic numbers that are relatively abelian over quadratic imaginary number fields.

And David Hilbert saying

The theory of complex multiplication is not only the most beautiful part of mathematics but also of the whole of science.

And the great mathematician David Mumford writing:

Especially, I became obsessed with a kind of passion flower in this garden, the moduli spaces of Riemann. I was always trying to find new angles from which I could see them better.

I wonder who this gmachine1728 is, and in case he sees this, he is welcome to use this contact page https://gmachine1729.wpcomstaging.com/%e8%81%94%e7%b3%bb-%d0%ba%d0%be%d0%bd%d1%82%d0%b0%d0%ba%d1%82-contact/.

Or email me at gmachine1729 at foxmail.com. I have had some interesting, high-quality people find me through this blog and contact me, and I also became good friends with some of them, chatting with them a fair bit on the internet. So yes, readers don’t be shy. Feel free to contact me if you have something worthwhile to talk about or something you want to ask. I like to think of myself as an easily approachable person. I will certainly respect your privacy and won’t leak your email information to anyone else without your permission.