Let $X$ denote a smooth projective curve defined over $\mathbb{Z}[1/N]$, and its base change $ \overline{X} $ to $ \overline{\mathbb{Q}} $. Let $ V $ be a $ p $-adic local system on $X$ ($p\mid N$), and by abuse of notation denote by $V$ also its pullback to $ \overline{X} $. I am particularly interested in the étale cohomology group $ H_{\text{ét}}^1(\overline{X}, V) $, and more precisely in the action of complex conjugation on this space.
Assume complex conjugation has equidimensional $\pm 1$ eigenspaces on $V$ (this would mean in particular that $V$ is even dimensional). Are the dimensions of the positive and negative eigenspaces of $ H_{\text{ét}}^1(\overline{X}, V)$ also equal? Any references or insights into this problem would be highly appreciated.
