On algebraic cycles on complex Abelian schemes over smooth projective curves

If the Hodge conjecture holds for some generic (in the sense of Weil) geometric fibre $X_s$ of an Abelian scheme $\pi\colon X\to C$ over a smooth projective curve $C$, then numerical equivalence of algebraic cycles on $X$ coincides with homological equivalence. The Hodge conjecture for all complex Abelian varieties is equivalent to the standard conjecture $B(X)$ of Lefschetz type on the algebraicity of the Hodge operator $\ast$ for all Abelian schemes $\pi\colon X\to C$ over smooth projective curves. We investigate some properties of the Gauss–Manin connection and Hodge bundles associated with Abelian schemes over smooth projective curves, with applications to the conjectures of Hodge and Tate.