Action of complex conjugation on etale cohomology

Question

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 recall reading somewhere that the representation is odd. Is this true more generally, for example when $g>1$?

If so, I would appreciate a reference for the statement. An intuitive explanation for why this is expected would also be appreciated.

Answer 1

The Weil pairing (or Poincaré duality in étale cohomology) gives a Galois-equivariant symplectic form $$H^1(\overline{X}, \mathbb Q_p) \times H^1(\overline{X}, \mathbb Q_p) \to \mathbb Q_\ell(-1).$$

The action of complex conjugation on $ Q_\ell(-1)$, the inverse of the cyclotomic character, is by multiplication by $-1$, since roots of unity lie on the unit circle and hence are sent to their own inverse by complex conjugation.

It follows that complex conjugation sends the symplectic form to minus the symplectic form. Complex conjugation also acts by a matrix of order $2$. Such matrices necessarily have eigenvalue $1$ with multiplicity $g$ and $-1$ with multiplicity $g$, since their $1$ and $-1$ eigenspaces must both be isotropic subspaces, hence both maximal. In particular their determinant is $(-1)^g$.