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Right, but the dodecahedron is a modular curve, and I'm guessing this is how Hermite came up with the solution to the quintic using modular functions in 1858. I
by mathgenius 9y ago
Right, but the dodecahedron is a modular curve, and I'm guessing this is how Hermite came up with the solution to the quintic using modular functions in 1858. If there are any real number theorists here they might be able to clarify this some more. Modular functions, modular forms, automorphic forms, these are all key ingredients of the Langlands program.
- goldenkey 9y agoI believe you're talking about the free algebras on groups of modular forms of a certain 'weight.' That's not exactly the modular forms themselves. Here's some more information: https://johncarlosbaez.wordpress.com/2017/12/31/quantum-mechanics-and-the-dodecahedron/ https://johncarlosbaez.wordpress.com/2017/12/31/quantum-mech...
- mathgenius 9y agoThat john baez article is about something else. Try this: https://arxiv.org/pdf/1308.0955.pdf https://arxiv.org/pdf/1308.0955.pdf
- throwaway080383 9y agoWhat do you mean when you say the dodecahedron is a modular curve? Is there some N for which X_1(N) can be naturally thought of as a dodecahedron or something? Edit: Okay glanced at the arxiv paper linked below. The dodecahedron is not a modular curve per se, but the link is as follows: The quotient X(5) -> X(1), which geometrically is a quotient from P^1 (the sphere) to itself, can also be thought of as quotienting the sphere by the group of symmetries of its inscribed dodecahedron. I believe this group can naturally be thought of as PSL_2(Z/5Z)
- paulpauper 9y agoIt's quite complicated (https://en.wikipedia.org/wiki/Bring_radical https://en.wikipedia.org/wiki/Bring_radical) ..amazing he derived it so long ago , long before the internet and modern research tools were invented. he had to scour for all that information and derive this solution.. even with the help of the internet I find it hard to understand