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If $E_1$ and $E_2$ are two elliptic curves over $\mathbb{Q}$ such that $|E_1(\mathbb{F}_p)|=|E_2(\mathbb{F}_p)|$ for all primes $p$, what does this tell us about the relationship between $E_1$ and $E_2$?

E.g. the identity holds (at least for all primes $p<500$) when $E_1$ and $E_2$ have the equations $y^2+y=x^3+x^2+2x+4$ and $y^2+y=x^3+x^2-208x-1256$. In what way are these curves similar?

They aren't isomorphic because their reduced forms ($y^2=x^3+ax+b$) are distinct. One of them has a torsion subgroup over $\mathbb{Q}$ of size 5, and the other has trivial torsion subgroup. The difference between the RHS's is $210x+1260=(2)(3)(5)(7)(x+6)$; is it relevant that this has a primorial factor? (Obviously it explains why $|E_1(\mathbb{F}_p)|=|E_2(\mathbb{F}_p)|$ for $p=2,3,5,7$, but why should this still hold for larger $p$ ??)

Many thanks for any help with this!

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    A little bird once told me "if and only if they are isogenous," but don't take my word for it.2012-09-02
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    @QiaochuYuan: thanks - I wondered if that might be the condition! :-) How can I prove that these specific curves are isogenous?2012-09-02
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    Ah, never mind; some Googling reveals that what I said is true at a fixed prime $p$ (apparently it is a theorem of Tate) but I don't know if it's true globally.2012-09-02
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    Are you skilled enough to factor out by the subgroup of order $5$? It’d be a useful exercise even if you didn’t get the other curve as the quotient.2012-09-02
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    The global theorem is apparently a theorem of Faltings (at least if I've got the hypotheses straight; I'm a little concerned about the primes of bad reduction).2012-09-02
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    @Lubin: I've found the torsion subgroup of order 5 for $E_1$, namely $\{0,(-1,1),(2,-5),(2,4),(-1,-2)\}$, but not sure what you mean by factoring out. Does this mean to find generators for the non-torsion part of $E_1(\mathbb{Q})$, and then see whether they generate $E_2(\mathbb{Q})$?2012-09-02
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    @QiaochuYuan: is this the theorem referred to here? http://w3.uwyo.edu/~chall14/papers/hall-perucca--radical-elliptic-curves.pdf It's fine if it only applies to primes of good reduction; I'm just interested in what the series looks like for primes of good reduction. (The $E_1$ and $E_2$ above have bad reduction at the same primes.)2012-09-02
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    @Harry: yes, that looks right.2012-09-02
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    Your elliptic curve $E$ is a group, and it has a finite subgroup $S$. Then $E/S$ is also an elliptic curve. The primitive way of finding $E/S$ is to get the subfield of its function field that’s fixed under the action of $S$, acting as translations of the underlying space. But Qiaochu may be able to tell you better methods.2012-09-02
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    You probably knew this, but these are the two curves c1 and c2 of level 75 in Cremona's tables. So one way to see that they are isogenous is just to look them up.2012-09-02
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    @JeremyTeitelbaum: thanks for this! Are these tables available online? (I found Cremona's page at Warwick, but it seems you need to download an enormous file to get any information, and I'm not sure what a conductor is.)2012-09-02
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    Harry, what prof. Lubin is suggesting is that you write down the formulas (using the addition rule on $E_1$) for what happens to a generic point $(x,y)$, when you add one of the five torsion points to it. This gives you five automorphisms of the function field $\mathbb{Q}(E_1)$, and if you can identify the fixed field of that automorphism group as the function field of $E_2$, then you have your isogeny.2012-09-02
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    @HarryMacpherson: the theorem of Faltings applies if you have equality for almost all primes. So you can skip to primes of good reduction for $E_1, E_2$.2012-09-03
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    You can easily check whether these curves are isogenous using Sage (www.sagemath.org). See this page http://www.sagemath.org/doc/reference/sage/schemes/elliptic_curves/ell_rational_field.html , in particular see the commands for isogeny_class, isogeny_graph and isogenies_prime_degree. Let me know if you have trouble using these commands. You can also browse Cremona's tables in Sage.2012-09-03
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    Regarding the Cremona Tables, I went here:http://homepages.warwick.ac.uk/~masgaj/ftp/data/ and downloaded the list of conductors up to 1000. Then I just searched the list for the interesting numbers in your equations: 208 and 1256. That found the curves right away.2012-09-03
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    @Jeremy: thanks, I've got it now. Just one more question: are Cremona's isogeny classes defined by $\mathbb{Q}$-isogeny or just isogeny? Because the theorem by Faltings needs $\mathbb{Q}$-isogeny!2012-09-03
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    Or are isogeny and $\mathbb{Q}$-isogeny the same for elliptic curves defined over $\mathbb{Q}$ ? This site seems to say they're not: http://seminariomatematico.dm.unito.it/rendiconti/cartaceo/53-4/389.pdf But I can't find anywhere else that defines $k$-isogeny as something different from isogeny!2012-09-03
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    @Harry: You are fishing in deep waters. In general two elliptic curves E and E' defined over a field k might be isogenous over an extension field but not over k. (In fact, E and E' might be isomorphic over an extension field but not over k). But to answer your narrower question, yes, Cremona's tables classify elliptic curves up to Q-isogeny.2012-09-03

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