Collection Rational Points on Fano and Similar Varieties
Organizer(s)
Date(s)
05/05/2024
Strong approximation for a family of norm varieties
By Fei Xu
Let $L/k$ be a finite extension of number fields and $q(t)$ be a polynomial over $k$. It is a classical problem to study weak approximation for the family of norm varieties defined by $$N_{L/k}(x) = q(t)$$ by various methods. The first non-trivial example for strong approximation with Brauer-Manin obstruction of the above equation was given by Derenthal and Wei, where $[L : k] = 4$ and $q(t)$ is an irreducible quadratic polynomial which has a root in $K$. In this talk, I will explain that the above equation satisfies strong approximation with Brauer-Manin obstruction under Schinzel’s hypothesis, where $L/k$ is cyclic, $q(t)$ is a product of distinct irreducible polynomials $q_i(t)$ with $1\leqslant i \leqslant n$ and one of the splitting fields $M_i$ of $q_i(t)$ having $M_i\cap L_{\text{ab}}\subset L$ with the maximal abelian extension $L_{ab}$ of $L$. This is a part of joint work with Cao and Wei.
