Poincaré institute

Reinventing rational points

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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.