When adding to the rational the $p$-torsion points $E[p]$ of an elliptic curve we obtain an extension containing the $p$-th roots of the unity, and whose Galois group can be embedded in $GL(2, \mathbb{F}_p)$. To what extent are such extensions coming from elliptic curves?

I mean, assume $K/\mathbb{Q}$ to be an extension whose Galois group can be embedded in $GL(2, \mathbb{F}_p)$ and containing the $p$-th roots of the unity (which is required to expect a positive answer), is $K$ obtained adding to $\mathbb{Q}$ the torsion points of some elliptic curve defined over the rationals?

Note that I'm not considering a particular Galois representation, but just Galois groups that can be embedded in some way into $GL(2, \mathbb{F}_p)$. Thanks!

  • 3
    $\begingroup$ In general there will be further conditions on the possible subgroups of $\mathrm{GL}_2(\mathbf{F}_p)$. First as you mention the determinant must be surjective onto $\mathbf{F}_p^{\times}$. It is also expected that for non CM ell. curves over Q and p big enough (independent of the curve), the Galois representation will be surjective onto $\mathrm{GL}_2(\mathbf{F}_p)$. A good starting point for this is Serre's article Propriétés des points d'ordre fini des courbes elliptiques, Inventiones 1972. $\endgroup$ Jul 4 '11 at 9:13
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    $\begingroup$ Once you fix an embedding into the general linear group, Serre's conjecture, now a theorem, implies that the representation arises from a modular form. Deligne's theory and the theory of congruences between modular forms shows that such representations indeed come from two dimensional torsion subgroups of the jacobians of the modular curves, but I think these don't have to come from an elliptic curve. $\endgroup$ Jul 4 '11 at 9:54
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    $\begingroup$ The $p=5$ case is misleading. The associated modular curve has genus 0 so if there are no local obstructions to there being points, then there are points. A more illuminating example would be $p$ some random 10-digit prime. The moduli space of elliptic curves giving rise to this Galois representation would be a curve of huge genus and there's no reason at all that it should have any rational points, so in general I don't think it's reasonable to expect that a Galois representation, even with cyclo det, should be the $p$-torsion of an ell curve. $\endgroup$ Jul 4 '11 at 22:24
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    $\begingroup$ Here's another cheap source of counterexamples: consider a reducible 2-dimensional Galois representation with cyclotomic determinant and a 1-dimensional subspace where Galois acts trivially. By Mazur this won't come from an ell curve if $p\geq 11$. $\endgroup$ Jul 4 '11 at 22:27
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    $\begingroup$ @Maurizio : Mazur's celebrated theorem tells us that (in the non CM case) for $p$ big enough, the image of Galois cannot be contained in a Borel subgroup of $\mathrm{GL}_2(\mathbf{F}_p)$. The bound on $p$ is uniform on $E$ and actually quite low (I think $p>37$ will do). See Modular curves and the Eisenstein ideal, Publ. Math. IHES 1977. $\endgroup$ Jul 5 '11 at 14:41

This question is discussed very carefully in Section 3 of the paper Mod $p$ representations on elliptic curves, by Frank Calegari (available here).

In particular, after noting that the answer is positive when $p \leq 5$ (as was already observed in the comments above), he proves (in Theorems 3.3 and 3.4) that if $p \geq 7$ then there exists a continuous representation $\rho: Gal(\overline{\mathbb Q}/ \mathbb Q) \to GL_2(\mathbb F_p)$ with cyclotomic determinant which does not come from the $p$-torsion of an elliptic curve. (He also notes that the same result was established by Dieulefait, in the paper Existence of non-elliptic mod $\ell$ Galois representations for every $\ell > 5$.)

  • $\begingroup$ Nice answer, i look forward to give a careful look at that paper soon. Thanks! $\endgroup$ Jul 14 '11 at 13:12

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