Using the modularity of elliptic curves is fine, and, if anything, makes a diophantine result more interesting. However, using the Classification of Finite Simple Groups leaves me feeling slightly dirty. Discuss.

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Using the modularity of elliptic curves is fine, and, if anything, makes a diophantine result more interesting. However, using the Classification of Finite Simple Groups leaves me feeling slightly dirty. Discuss.

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Hola profe, podria ser mas explicito? Como vincula la modularidad de curvas elipticas a la clasificacion de grupos finitos? Por que esta ultima no es de su agrado?

Roberto.

Hola Roberto,

No se vincula, es un contraste.

Harald

I had to use a diophantine result in my most recent paper. It left me feeling all dirty.

Are you just saying this as payback? But surely you are not from the finite-groups party? Or was there a particularly good reason why the use of that diophantine result was dirty?

Well, we’re using a really fancy theorem, and still this is the part that pulls down the final result (from a power bound to a exp(-cn/log log n) bound), and it doesn’t look like we could do better by using a fancier diophantine result, even in principle.

Not that I really know anything about diophantine number theory, anyway. I knew enough to suspect that some result like that might be true, and to ask around about it, but we could well be missing something basic.

That was supposed to be exp(-c log n / log log n).

But at least half of the diophantine statements I know can be considered as being applied harmonic analysis…

For me, there are many reasons; here are some “en vrac”:

* I already understand the strategy of the proof of modularity, and given a few months or a year of concentrated work, I could check for myself the full details of the proof; it’s clear this wouldn’t work for the classification.

* Believing in modularity is to believe that a certain list (of known-modular elliptic curvers) can be extended indefinitely, in a situation where we basically know how to add a new element concretely (if someone gives you a finite list, you create a j-invariant not in this list, and use the fact that there are algorithms to check modularity.) What needs to be done is to prove that this always works… In contrast, believing in classification is believing that a certain finite list is complete (the list of sporadic groups), when this list carries very little, by itself, to suggest it should be complete.

* Modularity is used often in an essential way; classification is often used for a very specific corollary that seems could/should be provable independently (e.g., the number of finite simple groups of a given order is at most 2…)

I think you’ve put it very well, thanks.