Let $E \subset \mathbb{P}_\mathbb{C}^2$ be an elliptic curve. If $E$ has complex multiplication (by anything) then the theory of complex multiplication in particular tells us that if $\sigma \in \textrm{Aut}(\mathbb{C})$ then $E^\sigma$ will be isogenous to $E$. Suppose $\textrm{End}(E) \cong \mathbb{Z}$ do we still have that $E^\sigma \cong E$ or are there some obvious counterexamples? What if we insist that the $j$-invariant, $j(E)$ is an algebraic number?
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We say that an elliptic curve $E$ over a number field $K$ is an elliptic $\mathbf{Q}$-curve if it is is isogenous to its Galois conjugates $E^\sigma$. These were first studied by Benedict Gross, but were later studied by Elkies, Ellenberg, Ribet, and many others. It's possible to show that $\mathbf{Q}$ curves are modular, independently of the BCDTW modularity theorem, and that even though they are defined over $K$, their $p$-torsion Galois representations (at least away from the degrees of said isogenies) descend down to $\mathbf{Q}$. Elkies also famously showed if $E_{/K}$ is a $\mathbf{Q}$-curve without CM, there must be a geometrically isogenous curve $E'_{/L}$ such that $Gal(L/\mathbf{Q}) \cong (\mathbf{Z}/2\mathbf{Z})^r$ for some integer $r$. This comes down to showing that the moduli space of elliptic $\mathbf{Q}$-curves of degree $N$ form a certain very special quotient of $X_0(N)$. For details on most of this, see Jordan Ellenberg's "$\mathbf{Q}$-curves and Galois representations" ( http://www.math.wisc.edu/~ellenber/MCAV.pdf ) |
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One way to get a counterexample is to take any elliptic curve $E$ over a quadratic field $K$ whose conductor is a prime ideal $\mathfrak{p}$ lying over a split prime $p$ in $\mathbb{Q}$. This means that $E$ has multiplicative reduction at $\mathfrak{p}$. If $\sigma$ is the non-trivial automorphism of $K/\mathbb{Q}$, then $E^\sigma$ has conductor $\mathfrak{p}^\sigma \neq \mathfrak{p}$, so $E^\sigma$ has good reduction at $\mathfrak{p}$. Any isogeny $E \to E^\sigma$ would be defined over a number field, and multiplicative/good reduction are both stable under finite field extension. Since isogenous curves have the same reduction type, $E$ and $E^\sigma$ cannot be isogenous. An example: the two curves of minimal norm conductor over $\mathbb{Q}(\sqrt{5})$ have norm conductor 31, but their conductors are the two different ideals of norm 31. Check out William Stein's table: http://modular.math.washington.edu/Tables/hmf/sqrt5/ellcurve_aplists.txt As a double check, you can even tell from the Hecke eigenvalues in the table that the curves are conjugate to one another. Maybe someone can give a simple counterexample (without using Elkies' theorem quoted by stankewicz) with integral $j$-invariant? |
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