I would like to know an example of a projective variety over a totally real field where a complex conjugation is not odd on some of its étale cohomology.

*Edit:* I am looking for the most interesting statement. Namely, is there an example of a connected projective variety $X$ of dimension $>0$ over a totally real field $F$ such that, for some $i\ne0,\dim X$, there is an irreducible constituent $\rho$ of the Galois representation $H^{i}_{et}(X_{\overline{\mathbb{Q}}},\overline{\mathbb{Q}}_{\ell})$ such that

- $\dim_{\overline{\mathbb{Q}}_{\ell}}\rho\ge2$,
- and for some complex conjugation $c$ in $\text{Gal}(\overline{F}/F)$, $|\text{Trace}(\rho(c))|>1$.

As posted in the comment, we know that it is expected to not arise from cohomological automorphic representations of $\text{GL}_{n}$, and it should have irregular Hodge–Tate weights.

irreducibleGalois representations (where it is the condition that the number of +1 and -1 eigenvalues for complex conj'n at each infinite place be equal if the dimension is even, or differ by 1 if the dimension is odd). E.g. would you consider the sum of 17 copies of the cyclotomic character to be odd? $\endgroup$ – David Loeffler Feb 5 at 9:04