Splitting of almost ordinary abelian surfaces in families and the 𝑆-integrality conjectures

Abstract Let 𝐴 be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic 𝑝. Suppose Δ is an infinite set of positive integers such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mfrac> <m:mi>m</m:mi> <m:mi>p</m:mi> </m:mfrac> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> (\frac{m}{p})=1 for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> </m:math> m\in\Delta . If 𝐴 does not admit any global real multiplication, we prove the existence of infinitely many places modulo which the reduction of 𝐴 has endomorphism ring containing <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> <m:mo>/</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi>x</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>−</m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathbb{Z}[x]/(x^{2}-m) for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> </m:math> m\in\Delta . This implies that there are infinitely many places modulo which 𝐴 is not simple, generalizing the main result of [D. Maulik, A. N. Shankar and Y. Tang, Reductions of abelian surfaces over global function fields, Compos. Math. 158 (2022), 4, 893–950] to the non-ordinary case. As another application, we also generalize the 𝑆-integrality theorem for elliptic curves over number fields, as proved in [M. Baker, S.-I. Ih and R. Rumely, A finiteness property of torsion points, Algebra Number Theory 2 (2008), 2, 217–248], to the setting of abelian surfaces over global function fields.