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

Mar 2023

This paper fills in all details in the announcement arXiv:2301.10054 on ourresults: we construct infinitely many non-isotrivial families of abelianvarieties of $\text{GL}_2$-type over four punctured projective lines with badreduction of type-$(1/2)_\infty$ via $p$-adic Hodge theory and Langlandscorrespondence. They lead to algebraic solutions of the Painleve VI equation.Recently Lin-Sheng-Wang proved the conjecture on the torsioness of zeros ofKodaira-Spencer maps of those type families. Based on their theorem we show theset of those type families of abelian varieties is {\sl exactly} parameterizedby torsion sections of the universal family of elliptic curves modulo theinvolution.

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