Hilbert’s 10th problem over number fields
SEMINARS, SEMINARS: NUMBER THEORY
We show that for every 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 is known to imply that Hilbert’s tenth problem over the ring of integers R of any number field has a negative solution. That is, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over R has solutions in R. In the pretalk, I’ll talk about CM abelian varieties and Selmer groups. This is joint work with Levent Alpöge, Manjul Bhargava, and Wei Ho.
Ari has also kindly agreed to give a pretalk from 2:00-2:45pm in SC 530. We especially encourage younger students to attend and ask questions!