Almost ordinary Abelian surfaces over global function fields with application to integral points

Let \(A\) be a non-isotrivial almost ordinary Abelian surface with possibly bad reductions over a global function field of odd characteristic \(p\). Suppose \(\Delta\) is an infinite set of positive integers, such that \(\left(\frac{m}{p}\right)=1\) for \(\forall m\in \Delta\). If \(A\) doesn't...

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Published in:arXiv.org 2023-04
Main Author: Jiang, Ruofan
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description Let \(A\) be a non-isotrivial almost ordinary Abelian surface with possibly bad reductions over a global function field of odd characteristic \(p\). Suppose \(\Delta\) is an infinite set of positive integers, such that \(\left(\frac{m}{p}\right)=1\) for \(\forall m\in \Delta\). If \(A\) doesn't admit any global real multiplication, we prove the existence of infinitely many places modulo which the reduction of \(A\) has endomorphism ring containing \(\mathbb{Z}[x]/(x^2-m)\) for some \(m\in \Delta\). This generalizes the \(S\)-integrality conjecture for elliptic curves over number fields, as proved in arXiv:math/0509485, to the setting of Abelian surfaces over global function fields. As a corollary, we show that there are infinitely many places modulo which \(A\) is not simple, generalizing the main result of arXiv:1812.11679 to the non-ordinary case.
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Number theory
title Almost ordinary Abelian surfaces over global function fields with application to integral points
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