On the determinant of an odd, continuous Galois representation. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T03:36:32Z http://mathoverflow.net/feeds/question/52553 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52553/on-the-determinant-of-an-odd-continuous-galois-representation On the determinant of an odd, continuous Galois representation. T.B. 2011-01-19T22:26:38Z 2011-01-21T17:32:12Z <p>In his paper, <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.dmj/1077305511" rel="nofollow">Duke paper</a>, Serre consider continuous, odd Galois representation $\rho: G_{\mathbb{Q}}\longrightarrow GL_{n}(\overline{\mathbb{F}}_{p})$ where $p$ is a rational prime. Roughly, (I don't understand much French except for the help from Google translation) Serre claims (section 1.3) that </p> <p>$\det(\rho(Frob_{l})) = \epsilon(Frob_{l})\omega^{k}(Frob_{l})$ for all prime $l\nmid pN$ where $N$ is defined as the level of the representation (with an explicit formula given in the paper) and $\epsilon$ is a Dirichlet character and $k$ is some positive integer.</p> <p>This seems to be standard since other papers cited it without reproving and I could not find any reference for the proof. In particular, my questions are:</p> <p>1) Where can I find a proof for this.</p> <p>2) What exactly is $\epsilon$, in some paper, there is the claim that $\epsilon$ is the unique quadratic character mod $p$ ramified only at $p$, and I do not understand where this comes from?</p> <p>3) How can one finds $k$.</p> <p>For motivation, I think $\det(\rho(Frob_{l}))$ is an important invariant to compute since, for example, it appears in the attachment equation that associates these representations with modular forms.</p> <p>Thanks in advance for any insight.</p> http://mathoverflow.net/questions/52553/on-the-determinant-of-an-odd-continuous-galois-representation/52558#52558 Answer by Felipe Voloch for On the determinant of an odd, continuous Galois representation. Felipe Voloch 2011-01-19T23:25:56Z 2011-01-21T17:32:12Z <p>$\det (\rho)$ is a one dimensional rep of the absolute Galois group of the rationals, i.e., it is a character. All such characters can be described by class field theory or, more simply, by the Kronecker-Weber theorem. So is a Dirichlet character and, by the hypotheses, its conductor divides $pN$. Factor it as a character of conductor $p$ (that will be $\epsilon$) times a character of conductor $N$. The latter is a power of the cyclotomic character and $k$ is defined to be that power. The bit about quadratic character must be under additional hypotheses.</p> <p>Edit: I got $p$ and $N$ switched above. The character of conductor $N$ is $\epsilon$. The character of conductor $p$ is a power of the cyclotomic character because $(\mathbb{Z}/p)^*$ is cyclic.</p>