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