Timeline for Shortest possible reasonably self-contained formulation of the modularity theorem
Current License: CC BY-SA 4.0
9 events
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| Dec 12, 2019 at 12:04 | comment | added | Bogdan Grechuk | Also, the way suggested by Xarles (via function fields), which I currently implemented at theorems.home.blog/2019/11/14/… , looks shorter - what's the main problem with it? Is this because you think that assuming familiarity of readers with the term "function field" is too much, and the definition therefore cannot be treated as a "reasonably self-contained"? | |
| Dec 12, 2019 at 12:01 | comment | added | Bogdan Grechuk | Dear François Brunault, thank you. The problem with Mathworld is that it talks about modular functions of level N, and "modular functions" is the link, which leads to mathworld.wolfram.com/ModularFunction.html , but the definition there is for level 1 only. Do you have a link or reference to a (reasonbly short and self-contained) definition of holomorphic modular functions of level N? | |
| Dec 10, 2019 at 22:06 | comment | added | François Brunault | The Mathworld article is correct except the obvious typo that the correct substitution is $x \to g(z)$. The most elementary way to express the existence of the map $X_0(N) \to E$ is the one given by Mathworld, namely that $E$ can be parametrised by modular functions of some level. It is then another (easier) theorem that the level can be taken to be the conductor of $E$. Moreover, this formulation avoids the introduction of modular forms, we only need holomorphic modular functions (to define these, we need to define "meromorphic at the cusps"). | |
| Nov 18, 2019 at 15:00 | comment | added | Bogdan Grechuk | Thank you everyone! David, this is exactly why I decided to double-check here. So, the second formulation does not work. So, I will use the first formulation with complex points instead of rational one, and with definition of modular elliptic curve suggested by Xarles.In case anyone interested, the corrected formulation is here theorems.home.blog/2019/11/14/… | |
| Nov 17, 2019 at 19:32 | comment | added | David Loeffler | In the second formulation you are muddling two slightly different interpretations of the word "modular"; any function of the form P(j(z))/Q(j(z)) will be modular of level 1. (This is wrong in the MathWorld article too.) | |
| Nov 15, 2019 at 21:25 | comment | added | Felipe Voloch | In the first formulation, curves are not determined by their rational points, so it's incorrect as written but can be fixed by, e.g. looking at complex points instead. | |
| Nov 15, 2019 at 20:53 | comment | added | Xarles | In the first formulation, you can say that the field of functions on $E$ (given by $\mathbb{Q}(x)[y]/(y^2-(x^3+ax+b)$ is contained in the field of functions of $X_0(N)$ for some $N$ (given by $\mathbb{Q}(x)[y]/(P_N(x,y))$). | |
| Nov 15, 2019 at 14:27 | history | edited | Bogdan Grechuk | CC BY-SA 4.0 |
added 995 characters in body
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| Nov 15, 2019 at 10:06 | history | asked | Bogdan Grechuk | CC BY-SA 4.0 |