# Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve

Edited after Noam Elkies' comment: From what I understand (very few actually), there exist elliptic curves defined over some number fields $\mathbb{K}$ Galois over $\mathbb{Q}$ which are isogenous to all their Galois conjugates and such elliptic curves are called $\mathbb{Q}$-curves.

Since two elliptic curves are isogenous if and only if they share the same L-function, can one define the automorphism group of the L-function corresponding to such an elliptic $\mathbb{Q}$-curve in such a way that this group is actually isomorphic to $\operatorname{Gal}(\mathbb{K}/\mathbb{Q})$?

EDIT November 25th 2013: it seems rather clear that the automorphism group of the isogeny class of the $\mathbb{Q}$-curve $E$ contains $\operatorname{Gal}(\mathbb{K}/\mathbb{Q})$ as a subgroup. As the L-function $F$ of $E$ determines its isogeny class entirely, the automorphism groups of $F$ and of the isogeny class of $E$ should be isomorphic. Therefore it remains to show that such a group doesn't contain any element not arising from $\operatorname{Gal}(\mathbb{K}/\mathbb{Q})$.

EDIT December 6th 2013: There may be a way to provide a definite answer to my question. Suppose $g_1, g_2, \cdots g_n$ are elements of the considered automorphism group not arising from $\operatorname{Gal}(\mathbb{K}/\mathbb{Q})$ and let $G$ be the group generated by $\operatorname{Gal}(\mathbb{K}/\mathbb{Q})\cup \{g_{1},\cdots g_{n}\}$. Then, provided $G$ is realizable over $\mathbb{Q}$, there should exist a number field $\mathbb{L}$ containing $\mathbb{K}$ as a proper subfield such that the considered $\mathbb{Q}$-curve is actually defined over $\mathbb{L}$.

Edit April 1rst 2015: I copy and paste here the content of a cross-post MSE question which, after getting one up vote, has received one down vote et two votes to close without any explanation even though I did my best to improve it. So here it is:

"If I'm not mistaken, the isogeny class of $E/K$ contains an abelian group $Z^{r_{alg}}$ which is the torsion free part of the Mordell-Weil group of $E/K$. According to the BSD conjecture, the quantities $r_{alg}$ and $r_{an}:=\min\{k, L^{(k)}_{E/K}(1)\neq 0\}$ should be equal.

What I would like to know is whether the set of sub $Z$-modules (taken in the increasing order of rank) of $Z^{r_{alg}}$ can be viewed as a Galois covering of $Z$, and equivalently if the set $\{L_{E/K}^{(k)}, L_{E/K}^{(k)}(1)=0\}$ (taken in the increasing order of derivation order) can be viewed as a Galois covering of $L_{E/K}$.

If so, are these two Galois coverings isomorphic with Galois group $\operatorname{Gal}(K/Q)$?

Of course, the objects considered in the present question are to be defined as sets endowed with a coherent topology to be viewed as genuine topological spaces. Perhaps the Zariski topology would do the job considering $\operatorname{spec} Z$ instead of $Z$, and the set of local factors of $L_{E/K}$ (i.e those that appear in the Euler product thereof) instead of $L_{E/K}$ itself.

The key idea is to somehow identify:

1) the group $Aut_{B}(X)$ of automorphisms of the covering $X$ of the base $B$ preserving $B$ (where $B$ stands for the topological space corresponding to $L_{E/K}$ and $X$ to the $r_{an}$ topological spaces corresponding to $L_{E/K}, L'_{E/K},...,L^{(r_{an}-1)}$ in this order) with $Aut(L_{E/K})$ which, as showed in the link above, is isomorphic to $\operatorname{Gal}(K/Q)$,

2) the group $Aut_{B'}(Y)$ of automorphisms of the covering $Y$ of the base $B'$ preserving $B'$ (where $B'$ stands for the topological space corresponding to $Z$ and $Y$ to the $r_{alg}$ topological spaces corresponding to $Z, Z^2,...,Z^{r_{alg}}$ in this order) with the automorphism group of the isogeny class of $E/K$ which, as showed in the link above, is isomorphic to $\operatorname{Gal}(K/Q)$."

If some people here ever vote to close, please give a good reason to do so.

Sorry, the premise is wrong: it is quite special to have $E^\sigma$ isogenous to $E$ for all $\sigma$, and most elliptic curves over a field other than $\mathbb Q$ do not satisfy this condition. (The $E$ that do satisfy it are called "$\mathbb Q$-curves".) – Noam D. Elkies Nov 24 '13 at 16:39
Thank you, but I think there's a misunderstanding. Maybe my phrasing was bad, but I meant to say that elliptic curves over $\mathbb{K}$ that are called $\mathbb{Q}$-curves are precisely those which are isogenous to all their Galois conjugates. I'll edit my question to improve my phrasing. – Sylvain JULIEN Nov 24 '13 at 16:48
You might start by looking at $\mathbb{Q}$-curves with CM, which were studied by Dick Gross in his thesis, published as: MR0563921 Gross, Benedict H. Arithmetic on elliptic curves with complex multiplication. Lecture Notes in Mathematics, 776. Springer, Berlin, 1980. It also contains a bunch of background material on $\mathbb{Q}$-curves. – Joe Silverman Nov 24 '13 at 17:13