Image of a Galois representation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T09:02:11Z http://mathoverflow.net/feeds/question/86179 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86179/image-of-a-galois-representation Image of a Galois representation Srilakshmi 2012-01-20T07:35:45Z 2013-04-03T11:27:51Z <p>Notation:</p> <ul> <li>$E$ is a non-CM Elliptic curve over $\mathbb{Q}$.</li> <li>$p$ is an ordinary prime.</li> <li>$f$ - cuspidal eigenform of weight $k$ = 2 attached with $E$.</li> <li>$\rho_f$ - the global 2-dimensional $p$-adic Galois representation attached with $f$. $\rho_f$ : $G_S$ $\rightarrow$ $\mathrm{GL}_2({\mathbb{Z}}_p)$.</li> <li>$G_S:= \mathrm{Gal}(\mathbb{Q}_S/{\mathbb{Q}})$, where $\mathbb{Q}_S$ - maximal unramified extension outside the set <code>$S=\{\text{ bad primes of } E \} \cup\{ p, \infty \}$</code>.</li> </ul> <p>Assume that the residual representation $\overline{\rho}_f$ is $p$-split. </p> <p>The prime $p$ is an ordinary prime. So the the image of $\rho_f$ restricted to the decomposition group $G_p:=\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ will be of the form $\rho_f$ $\mid$ $G_p$ $\sim$ <code>$\begin{pmatrix} a &amp; * \\ 0 &amp; d \end{pmatrix}$</code>. The residual representation $\overline{\rho}_f$ is $p$-split. So, $\overline{\rho}_f$ $\sim$ <code>$\begin{pmatrix} \omega \lambda_p^{-1}(\overline{a}_p) &amp; 0 \\ 0 &amp; \lambda_p(\overline{a}_p) \end{pmatrix}$</code>, where $\lambda_p$ is an unramified character which sends $\mathrm{Frob}_p$ to $\overline{a}_p$, $\overline{a}_p \in \mathbb{F}_p$ is the mod $p$ reduction of the $p$-th coefficent $a_p$ of $f$, and $\omega$ is the $p$-adic cyclotomic character.</p> <blockquote> <p><strong>Question</strong>: What is the image of the representation $\rho_f: G_p \rightarrow \mathrm{GL}_2(\mathbb{Z}/p^n \mathbb{Z})$, where $(\mathbb{Z}/p^n\mathbb{Z})^2 \simeq (E[p^n])$, the $p^n$-torsion points of $E$, for some fixed $n \geq 2$? Is it possible to compute it (or atleast it's order) by using MAGMA/SAGE/PARI?</p> </blockquote> http://mathoverflow.net/questions/86179/image-of-a-galois-representation/86195#86195 Answer by SGP for Image of a Galois representation SGP 2012-01-20T10:44:40Z 2012-01-20T10:44:40Z <p>See <a href="http://www.digizeitschriften.de/en/dms/toc/?PPN=GDZPPN002089629" rel="nofollow">Serre's paper</a> where he shows how the the image in $GL_2(Z_p)$ (and hence mod $p^n$ for any $n>0$) is determined by the image of Galois in $GL_2(Z/{pZ})$; there are more recent works by Zywina among others.treating the case of abelian varieties. Serre's paper does the case of non-CM elliptic curve (the CM case can be found in Serre-Tate's Good reduction paper).</p> http://mathoverflow.net/questions/86179/image-of-a-galois-representation/126300#126300 Answer by unramified for Image of a Galois representation unramified 2013-04-02T17:31:58Z 2013-04-03T11:27:51Z <p>One can say something about the image of $\rho_f|G_p$ by checking if $f$ has a companion form mod $p^n$. This can be explicitly done because one knows what the weight of this companion (if it exists) should be ($p^{n-1}(p-1)$, since $k=2$) and the conguences mod $p^n$ that the form's Fourier coefficients must satisfy <em>vis a vis</em> the $a_p$'s. One need only check that these congruences are satisfied up to the Sturm bound to conclude that the companion exists. If a companion mod $p^n$ exists then $\rho_f|G_p$ mod $p^n$ splits and $p^n$ won't divide the order of its image, else it will.</p>