TY - JOUR
T1 - Rank stability in quadratic extensions and Hilbert’s tenth problem for the ring of integers of a number field
AU - Alpöge, Levent
AU - Bhargava, Manjul
AU - Ho, Wei
AU - Shnidman, Ari
N1 - Publisher Copyright:
© The Author(s) 2025.
PY - 2025
Y1 - 2025
N2 - We show that for any quadratic extension of number fields K/F, there exists an abelian variety A/F of positive rank whose rank does not grow upon base change to K. 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 OK of integers of any number field K, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over OK has solutions in OK.
AB - We show that for any quadratic extension of number fields K/F, there exists an abelian variety A/F of positive rank whose rank does not grow upon base change to K. 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 OK of integers of any number field K, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over OK has solutions in OK.
UR - https://www.scopus.com/pages/publications/105023643898
U2 - 10.1007/s00222-025-01392-3
DO - 10.1007/s00222-025-01392-3
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AN - SCOPUS:105023643898
SN - 0020-9910
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
ER -