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Let

$ E $ - Elliptic curve defined over $ {\mathbb{Q}} $.

$G_{\mathbb{Q}}$ - The absolute Galois group, $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) $ of $\mathbb{Q}$.

$ E[3] $ - $3$-torsion points of $ E $.

Suppose $ \rho$ denotes the $ G_{\mathbb{Q}} $-representation associated to $ E[3]$. If $ E $ has a $3$-torsion point over $ \mathbb{Q} $, how to prove $$ \rho \sim \left( {\begin{array}{cc} 1 & \eta \\ 0 & \chi \\ \end{array} } \right) $$ where $ \chi $ is the mod 3 cyclotomic character $?$

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1 Answer 1

up vote 7 down vote accepted

Assuming that $E$ has a rational $3$-torsion point, that point must be fixed, and hence your representation takes the form $$ \rho \sim \left( \begin{matrix} 1 & \eta_{i} \\ 0 & \chi \end{matrix}\right)$$ as you say. It's a general fact that the determinant of the $G_{\mathbb{Q}}$ representation attached to an elliptic curve is the cyclotomic character. This follows because the Weil pairing is bilinear, alternating, and Galois-invariant. (See Proposition III.8.1 in Silverman's Arithmetic of Elliptic Curves.)

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If $E$ has no rational $3$-torsion point, will $\rho$ still have the same representation $?$ –  Robert May 22 at 17:14
    
@Robert No, not necessarily. –  Olivier May 22 at 17:47
1  
@Robert No, never. –  Joël May 22 at 19:28

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