A natural mod-n representation of the automorphisms of an elliptic curve - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:50:26Z http://mathoverflow.net/feeds/question/87539 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87539/a-natural-mod-n-representation-of-the-automorphisms-of-an-elliptic-curve A natural mod-n representation of the automorphisms of an elliptic curve Vesna Stojanoska 2012-02-04T17:30:07Z 2012-02-04T17:30:07Z <p>Let $E$ be an elliptic curve over a field of characteristic $p$, and let $n$ be an integer coprime to $p$. Then $E[n]$, the kernel of multiplication by $n$ on $E$, is (etale-locally) isomorphic to $(\mathbb{Z}/n)^2$. A level-$n$-structure on $E$ is a specification of such an isomorphism $(\mathbb{Z}/n)^2\to E[n]$.</p> <p>Now $GL_2(\mathbb{Z}/n)$ acts on the set of level structures by pre-composition, and even more, the set of level-$n$-structures on $E$ is a $GL_2(\mathbb{Z}/n)$-torsor over the point on the moduli stack of elliptic curves determined by $E$ (I mean here the stacky point $B Aut(E)$).</p> <p>The group $G:=Aut(E)$ acts on the set of level structures on $E$ by post-composition. This determines a morphism $\rho: G \to GL_2(Z/n)$. Is this morphism explicitly known in general? </p> <p>I can describe this representation in the specific cases of $p=2$ and $n=3$, or $p=3$ and $n=2$, by writing down explicit equations for the curve and knowing explicitly in terms of those equations what do choices of points of the given order look like. But these are presumably more complicated cases in the sense that the structure of $G$ can be more complicated at $p=2$ or $3$.</p> <p>However, if the characteristic is not $2$ or $3$, $G$ is isomorphic to $\mu_k$, where $k$ is $2,4$ or $6$, depending on the $j$-invariant. One could write down representations of these $\mu_k$ in $GL_2(\mathbb{Z})$ that descend to representations in $GL_2(\mathbb{Z}/n)$ for all $n$. (One condition we probably need is that $-1\in G$ maps to the identity matrix.) For example, $\left(\begin{smallmatrix} -1 &amp; -1 \\ 1 &amp; 0 \end{smallmatrix}\right)$ has order $3$, and $\pm\left(\begin{smallmatrix} 0 &amp; 1 \\ 1 &amp; 0 \end{smallmatrix}\right)$ has order 2. How can I know if (or when) these matrices determine the representations I am looking for?</p> <p>Thanks!</p>