The purpose of this note is introducing a method for proving the existence of no rational points on a coarse moduli space $X$ of abelian varieties over a given number field $K$, in cases where the moduli problem is not fine and points in $X(K)$ may not be represented by an abelian variety (with additional structure) admitting a model over the field $K$. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired on the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that to a point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided $\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$. We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary quadratic field or an Atkin-Lehner quotient over $\mathbb{Q}$.

Darmon cycles are a higher weight analogue of Stark--Heegner points. They yield local cohomology classes in the Deligne representation associated with a cuspidal form on $\Gamma _{0}( N) $ of even weight $k_{0}\geq 2$. They are conjectured to be the restriction of global cohomology classes in the Bloch--Kato Selmer group defined over narrow ring class fields attached to a real quadratic field. We show that suitable linear combinations of them obtained by genus characters satisfy these conjectures. We also prove $p$-adic Gross--Zagier type formulas, relating the derivatives of $p$-adic $L$-functions of the weight variable attached to imaginary (resp. real) quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express the second derivative of the Mazur--Kitagawa $p$-adic $L$-function of the weight variable in terms of a global cycle defined over a quadratic extension of $\mathbb{Q}$.