On the standard conjecture for complex Abelian schemes over smooth projective curves

We reduce the Hodge conjecture for Abelian varieties to the question of the existence of an algebraic isomorphism $H^2(C,R^{2d-i}\pi_\ast\mathbb Q)\widetilde\rightarrow, H^0(C,R^i\pi_\ast\mathbb Q)$ for all $i\geqslant 2$ and all principally polarized complex Abelian schemes $\pi\colon X\to C$ of relative dimension $d$ over smooth projective curves. If the canonically defined Hodge cycles $\alpha_i(X/C)\in H^0(C,R^i\pi_\ast\mathbb Q)\otimes H^0(C,R^i\pi_\ast\mathbb Q)$ are algebraic for all integers $i\geqslant 2$, then the Grothendieck standard conjecture $B(X)$ on the algebraicity of the operators $\Lambda$ and $\ast$ holds for $X$. We prove $B(X)$ for an Abelian scheme under the assumption that $\operatorname{End}(X_s)=\mathbb Z$ for some geometric fibre $X_s$ of non-exceptional dimension.