Rank stability in quadratic extensions and Hilbert’s tenth problem for the ring of integers of a number field

Abstract We show that for any quadratic extension of number fields $K/F$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:math> , there exists an abelian variety $A/F$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:math> of positive rank whose rank does not grow upon base change to $K$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> . This result implies that Hilbert’s tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring ${\mathcal{O}}_{K}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> of integers of any number field $K$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> , there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over ${\mathcal{O}}_{K}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> has solutions in ${\mathcal{O}}_{K}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> .