Let $X$ be a genus $g$ smooth projective curve, defined over $\mathbb{Q}$, and let $\overline{X}$ denote the base change of $X$ to $\overline{\mathbb{Q}}$. It is well known that $H^1_{\text{ét}}(\overline{X}, \mathbb{Q}_p)$ has the structure of a $p$-adic Galois representation. When $g=1$, I reca...