On 𝜓-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a semiabelian variety over an algebraically closed field of arbitrary characteristic, endowed with a finite morphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi colon upper A right-arrow upper A"> <mml:semantics> <mml:mrow> <mml:mi> ψ </mml:mi> <mml:mo>:</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\psi : A\to A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper, we give an essentially complete classification of all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi"> <mml:semantics> <mml:mi> ψ </mml:mi> <mml:annotation encoding="application/x-tex">\psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant subvarieties of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For example, under some mild assumptions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A comma psi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi> ψ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(A,\psi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we prove that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi"> <mml:semantics> <mml:mi> ψ </mml:mi> <mml:annotation encoding="application/x-tex">\psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant subvariety is a finite union of translates of semiabelian subvarieties. This result is then used to prove the Manin-Mumford conjecture in arbitrary characteristic and in full generality. Previously, it had been known only for the group of torsion points of order prime to the characteristic of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The proofs involve only algebraic geometry, though scheme theory and some arithmetic arguments cannot be avoided.