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# Strong approximation for a family of norm varieties

## 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 $$L_{\text{ab}}$$ of $$L$$. This is a part of joint work with Cao and Wei.