The $\mathrm{\ell }$-parity conjecture over the constant quadratic extension

For a prime $\mathrm{\ell }$ and an abelian variety $A$ over a global field $K$, the $\mathrm{\ell }$-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton-Dyer, the ${\mathbb{Z}}_{\mathrm{\ell }}$-corank of the ${\mathrm{\ell }}^{\mathrm{\infty }}$-Selmer group and the analytic rank agree modulo $2$. Assuming that $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}K>0$, we prove that the $\mathrm{\ell }$-parity conjecture holds for the base change of $A$ to the constant quadratic extension if $\mathrm{\ell }$ is odd, coprime to $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}K$, and does not divide the degree of every polarization of $A$. The techniques involved in the proof include the '{e}tale cohomological interpretation of Selmer groups, the Grothendieck-Ogg-Shafarevich formula, and the study of the behavior of local root numbers in unramified extensions.