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If I understand FC's remark under the post "Very strong multiplicity one for Hecke eigenforms," in the course of Faltings's proof of the Tate conjecture, Faltings proves the following statement: let E/Q and F/Q be elliptic curves and write Q(E[p]) (resp. Q(F[p])) for the number field obtained by adjoining the x and y coordinates of the p-torsion of E to Q. Then if Q(E[p]) = Q(F[p]) for infinitely many primes p, E/Q and F/Q are isogenous.

Learning of this result prompted me to wonder: suppose P is a finite set of primes. Then do there exist E/Q and F/Q such that Q(E[p]) = Q(F[p]) for each p in P with E/Q and F/Q not isogenous?

If not in general, what is known about the particular P for which the above question does (or doesn't) have an affirmative answer?

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Dear FC, I think that you may misinterpret where my question was coming from. I don't know arithmetic algebraic geometry and don't suppose learning an receiving an answer to my question (or many such questions)) serves as a substitute for learning arithmetic algebraic geometry. So why did I ask the question that I did (and why do I ask many questions of a similar flavor)? Because my learning style is such that I find it very difficult to follow an axiomatic presentation of material, whereas concrete and surprising details sticks in my mind. –  Jonah Sinick Oct 31 '09 at 18:54
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As such, if I am going to motivate myself to eventually learn arithmetic geometry, it won't be by reading through hundreds of pages of Bourbaki-esque exposition from start to finish but by learning a variety of facts that capture my interest at my current level of understanding, then by learning the things that I need to to understand them, then by learning the things that I need to to understand these things, etc. That is, I have a strong preference for starting at the level of surprising phenomena and working backward toward learning what I need to to understand. –  Jonah Sinick Oct 31 '09 at 18:59
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The fact that you mentioned falling out of Faltings work was quite interesting to me (thanks for mentioning it) and I immediately wanted to place it into context which is what prompted me to ask this question. Now, your doubt as to whether I had thought very much about my initial question is well founded - e.g. thinking about the case where E[p] has a rational P-torsion point would be within my power and I didn't take the time to do so. But doing so would take some effort for me and would be effortless some others (some of whom would appreciate the chance to articulate their understanding). –  Jonah Sinick Oct 31 '09 at 19:06
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If you personally do not wish to indulge me I wouldn't fault you - I know that you have a lot on your plate. But I think that for a certain kind of person such as myself, asking and answering questions of the type that mine falls into is a productive activity. –  Jonah Sinick Oct 31 '09 at 19:17
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I think this is an appropriate question and disagree with FC. –  David Zureick-Brown Oct 31 '09 at 19:25
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1 Answer

up vote 7 down vote accepted

Most people would say no. Indeed, there's a conjecture, most commonly ascribed to Frey, that for p a SINGLE large enough prime, E[p] isomorphic to F[p] implies that E and F are isogenous. I believe p=37 is thought to be large enough, but don't hold me to it.

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Thanks Jordan. (This is the sort of repsonse that I was looking for). –  Jonah Sinick Oct 31 '09 at 19:46
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