Timeline for Why no abelian varieties over Z?
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| Mar 25, 2015 at 5:03 | comment | added | Filippo Alberto Edoardo | @ Emerton: but oddly enough, Khare-Wintenberger need an inductive argument whose basic step relies upon Schoof's work on abelian varieties over number field with few bad places... ;) | |
| Feb 10, 2010 at 6:07 | history | edited | Emerton | CC BY-SA 2.5 |
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| Jan 6, 2010 at 11:56 | vote | accept | Ilya Nikokoshev | ||
| Jan 6, 2010 at 6:02 | comment | added | Emerton | I should also add, regarding my paranthetical remark, that Tate's non-existence result for mod 2 2-dim'l Galois reps. unram. outside 2 is a crucial ingredient in the proof of Serre's conjecture by Khare and Wintenberger, so logically that situation is quite different from the elliptic curve case, where Tate's non-existence result plays no role in the proof of modularity. | |
| Jan 6, 2010 at 5:51 | comment | added | S. Carnahan♦ | Oh, that's really good to know. Thanks! | |
| Jan 6, 2010 at 5:17 | comment | added | Emerton | I don't think there is any hidden circular reasoning there, and in fact is a perfectly good way to think about it. Actually, Tate's result served as an early consistency check for the modularity conjecture. (Just as his result on the non-existence of continuous representations $\rho:G_{\mathbb Q} \to GL_2(\bar{F}_2)$ unramified outside 2 served as an early consistency check on Serre's conjecture.) | |
| Jan 6, 2010 at 5:13 | comment | added | S. Carnahan♦ | One might say that the elliptic curve case follows from nonexistence of weight 2 level 1 cusp forms + modularity, but there may be some hidden circular reasoning (and it seems like a backwards way to look at things). | |
| Jan 6, 2010 at 1:23 | comment | added | Anweshi | @Ilya. I took a look at page 16. That is Chapter 2 on Faltings theorem, and I had imagined that it was a separate article. Read on, you will find that Faltings settled that too. The role of Shafarevich is that he proved it for elliptic curves -- the proof is rather easy, it can be found in Silverman's book on elliptic curves. The higher dimensional case was then called "Shafarevich conjecture". | |
| Jan 6, 2010 at 0:23 | comment | added | Emerton | From a number theorist's point of view, $p$-adic Hodge theory is one of the key ingredients in the theory of motives, so these arguments are motivic, in a certain sense. (Perhaps one can say that $p$-adic Hodge theory encodes arithmetic properties of motives in a way analogously to the way that Hodge theory encodes geometric and analytic properties.) | |
| Jan 6, 2010 at 0:16 | history | edited | Emerton | CC BY-SA 2.5 |
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| Jan 5, 2010 at 23:50 | comment | added | Anweshi | Faltings proved that also, along with the Mordell conjecture. See Darmon's article on Faltings' theorem. | |
| Jan 5, 2010 at 23:33 | comment | added | Ilya Nikokoshev | Re: generalized by Faltings to abelian varieties. That's one more reason I ask: Darmon seems to call this a Shafarevich Conjecture (page 16) rather then a result. | |
| Jan 5, 2010 at 23:23 | history | edited | Emerton | CC BY-SA 2.5 |
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| Jan 5, 2010 at 23:16 | history | answered | Emerton | CC BY-SA 2.5 |