Timeline for When are K-automorphisms of the n-torsion of an elliptic curve E/K liftable to K-endomorphisms of E?
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| Nov 17, 2013 at 10:53 | comment | added | Yuri Zarhin | It seems to me that for elliptic curves the question (for non-CM) elliptic curves $E$ about the Galois image in $Aut(E[p])$ was raised by Serre in his (already mentioned) 1972 Inventiones paper. Some partial (but very important) results in this direction were obtained recently by Yuri Bilu and Pierre Parent annals.math.princeton.edu/2011/173-1/p13 . | |
| Nov 8, 2013 at 12:24 | comment | added | Stefan Keil | And is there a reference where it is conjectured for $d=1$? | |
| Nov 8, 2013 at 11:10 | comment | added | Yuri Zarhin | Dear Stefan, you are welcome. As far as I know, nobody stated it as a conjecture for $d>1$. | |
| Nov 8, 2013 at 10:05 | comment | added | Stefan Keil | Dear Yuri, thanks a lot for your answer and your reference. Remark 5.4.7 in your Inv. Math. paper is exactely what I was looking for. I used the following version: Fix an abelian variety A/K. Then there is a finite set S=S(A,K) of rational primes, such that for all primes powers p^k, with p is not in S, every Galois equivariant endomorphism A[p^k]->A[p^k] is induced by an endomorphism of A. I'd like to reask my third question. Is it conjectured that there is a 'global' finite set S=(d,K) of rational primes, with d a positive integer, such that the above holds for all A/K, such that dim(A)=d? | |
| Nov 8, 2013 at 9:56 | vote | accept | Stefan Keil | ||
| Nov 7, 2013 at 17:52 | history | answered | Yuri Zarhin | CC BY-SA 3.0 |