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we construct infinitely many non-isotrivial families of abelian varieties of$GL_2$-type over four punctured projective lines with bad reduction oftype-$(1/2)_\infty$ via $p$-adic Hodge theory and Langlands correspondence.They lead to algebraic solutions of Painleve VI equation. RecentlyLin-Sheng-Wang proved the conjecture on the torsionness of zeros ofKodaira-Spencer maps of those type families. Based on their theorem we show theset of those type families of abelian varieties is exactly parameterized bytorsion sections of the universal family of elliptic curves modulo theinvolution. After our paper submitted in arxiv, Lam-Litt gave a totally newconstruction of those abelian schemes by applying Katz's middle convolution.

Constructing families of abelian varieties of $\text{GL}_2$-type over  $4$-punctured complex projective line via $p$-adic Hodge theory and Langlands  correspondence and application to algebraic solutions of Painleve VI equation