I'm wondering about the notion of "oddness" in the theory of Galois representations. Usually, a Galois representation $$\rho : G_\mathbb{Q}\rightarrow GL_2(\mathbb{Q}_\ell)$$ is said to be odd if the image of complex conjugation $c\in G_\mathbb{Q}$ has determinant $-1$.

In particular, I'm interested in the case where $\rho = \rho_{E,\ell}$ is the Galois representation on the Tate module $T_\ell(E)$ of an elliptic curve $E/\mathbb{Q}$.

In this case, this fact is "proven" in Diamond-Shurman's A First Course in Modular Forms, but their proof goes by way of proving the statement for Galois representations associated to modular forms, and then using modularity to get the result for Galois reps on elliptic curves over $\mathbb{Q}$.

My questions are:

  1. For an elliptic curve over $\mathbb{Q}$, does $\rho_{E,\ell}(c)$ have characteristic polynomial $x^2-1$?

  2. If so, does (1) hold for any elliptic curve over $\mathbb{R}$?

References would be much appreciated.

  • 7
    $\begingroup$ The determinant is the $\ell$-adic cyclotomic character, which has value $-1$ on complex conjugation since the complex conjugate of any root of unity is its reciprocal. Since a $2 \times 2$ matrix of order 2 in characteristic 0 with determinant $-1$ must have eigenvalues $1$ and $-1$, this settles your questions for elliptic curves over any field equipped with an embedding into $\mathbf{R}$, by entirely elementary means (no need for modular forms, etc.). $\endgroup$ – nfdc23 Oct 9 '16 at 6:12
  • $\begingroup$ Ah fantastic! if you want to just copy/paste your comment into an answer I'd be happy to accept it. $\endgroup$ – stupid_question_bot Oct 9 '16 at 6:50
  • $\begingroup$ @rtz Hi I stopped Diamond's book at the algebraic geometry chapters, but I studied the analytic chapters, and I have no problems with L-functions. Can you tell me in a few words what is a Galois representation associated to a modular form ? (possibly an example ?) $\endgroup$ – reuns Oct 9 '16 at 9:57
  • $\begingroup$ @user1952009 If you have a weight 2 Hecke eigenform $f$ for some congruence subgroup $\Gamma$, you can think of it as a holomorphic differential 1-form $\omega_f$ on the (compact) genus $g$ modular curve $X(\Gamma)$. Since $f$ is an eigenform, the Hecke operators stabilize the vector space generated by $f$, and since they descend to an action on the Jacobian $J(\Gamma)$ (which we think of as $\Omega^1_{hol}(X)^{\wedge}/H_1(X,\mathbb{Z})$), where they cut out a sub abelian variety of dimension equal to $[K_f,\mathbb{Q}]$, where $K_f$ is the field generated by the Fourier coefficients of $f$. $\endgroup$ – Will Chen Oct 9 '16 at 18:05
  • $\begingroup$ @user1952009 If $T_\mathbb{Z}$ is the Hecke algebra (the algebra over $\mathbb{Z}$ generated by $T_n,\langle n\rangle$ for $n\ge 1$), then for every Hecke operator $T$, let $\lambda_f(T)$ be the eigenvalue of $f$ w.r.t. $T$. Let $I_f := Ker\lambda_f$, then the sub-abelian variety mentioned above is just $J_f := J(\Gamma)/I_fJ(\Gamma)$. In the above, we usually take $\Gamma = \Gamma_1(N)$ or $\Gamma_0(N)$, such that the corresp. modular curve is defined over $\mathbb{Q}$. Finally, the Galois rep assoc. to $f$ is just the representation of $G_\mathbb{Q}$ on the Tate module of $J_f$. $\endgroup$ – Will Chen Oct 9 '16 at 18:08

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