On the Hodge, Tate and Mumford–Tate conjectures for fibre products of families of regular surfaces with geometric genus 1

The Hodge, Tate and Mumford–Tate conjectures are proved for the fibre product of two non-isotrivial 1-parameter families of regular surfaces with geometric genus 1 under some conditions on degenerated fibres, the ranks of the Néron–Severi groups of generic geometric fibres and representations of Hodge groups in transcendental parts of rational cohomology.
Let $\pi_i:X_i\to C\quad (i = 1, 2)$ be a projective non-isotrivial family (possibly with degeneracies) over a smooth projective curve $C$. Assume that the discriminant loci $\Delta_i=\{\delta\in C \vert \mathrm{Sing}(X_{i\delta})\neq\varnothing\} \quad (i = 1, 2)$ are disjoint, $h^{2,0}(X_{ks})=1,\quad h^{1,0}(X_{ks}) = 0$ for any smooth fibre $X_{ks}$, and the following conditions hold:
$(i)$ for any point $\delta \in \Delta_i$ and the Picard–Lefschetz transformation $ \gamma \in \mathrm{GL}(H^2 (X_{is}, \mathbb{Q})) $, associated with a smooth part $\pi'_i: X'_i\to C\setminus\Delta_i$ of the morphism $\pi_i$ and with a loop around the point $\delta \in C$, we have $(\log(\gamma))^2\neq0$;
$(ii)$ the variety $X_i (i = 1, 2)$, the curve $C$ and the structure morphisms $\pi_i:X_i\to C$ are defined over a finitely generated subfield $k \hookrightarrow \mathbb{C}$.
If for generic geometric fibres $X_{1s}$ and $X_{2s}$ at least one of the following conditions holds:
$(a)$ $b_2(X_{1s})-{\mathrm{rank}}\, {\mathrm{NS}}(X_{1s})$ is an odd prime number, $\quad $ $b_2(X_{1s})-{\mathrm{rank}}\,{\mathrm{NS}}(X_{1s})\neq b_2(X_{2s})-{\mathrm{rank}} \,{\mathrm{NS}}(X_{2s})$;
$(b)$ the ring ${\mathrm{End}}_{\mathrm{Hg}(X_{1s})} {\mathrm{NS}}_{\mathbb{Q}}(X_{1s})^\perp$ is an imaginary quadratic field, $\quad b_2(X_{1s})-{\mathrm{rank}}\,{\mathrm{NS}}(X_{1s})\neq 4$,
${\mathrm{End}}_{\mathrm{Hg}(X_{2s})} {\mathrm{NS}}_{\mathbb{Q}}(X_{2s})^\perp$ is a totally real field or $b_2(X_{1s})-{\mathrm{rank}}\,{\mathrm{NS}}(X_{1s}) > b_2(X_{2s})-{\mathrm{rank}}\, {\mathrm{NS}}(X_{2s})$;
$(c)$ $[b_2(X_{1s})-{\mathrm{rank}}\,{\mathrm{NS}}(X_{1s})\neq 4, {\mathrm{End}}_{\mathrm{Hg}(X_{1s})} {\mathrm{NS}}_{\mathbb{Q}}(X_{1s})^\perp= \mathbb{Q}$; $b_2(X_{1s})-{\mathrm{rank}}\,{\mathrm{NS}}(X_{1s})\neq b_2(X_{2s})-{\mathrm{rank}} \,{\mathrm{NS}}(X_{2s})$, then for the fibre product $X_1 \times_C X_2$ the Hodge conjecture is true, for any smooth projective $k$-variety $X_0$ with the condition $X_1 \times_C X_2$ $\widetilde{\rightarrow}$ $X_0 \otimes_k \mathbb{C}$ the Tate conjecture on algebraic cycles and the Mumford–Tate conjecture for cohomology of even degree are true.