arXiv subjects
Publications and source records attributed to Dasheng Wei.
Let $X$ be a smooth toric variety over a number field $k$ such that the global sections of $X$ are constant. We show that $U(k)$ is dense in $U(\mathbf A_k)^{\text{Br}_1 (U)}$ for any open subset $U$ of $X$ such that the codimension of $X\setminus U$ in $X$ is at least 2.
We study strong approximation for the intersection of two affine quadrics. As its application, we prove the fibration method for weak approximation over number fields of rank four with nonsplit fibers split by quadratic extensions.
For an algebraic torus $S$, Blinstein and Merkurjev have given an estimate of $3$-th unramified cohomology $\bar{H}^3_{nr}(F(S),\mathbb Q/\mathbb Z(2))$ obtained from a flasque resolution of $S$. Based on their work, for the norm one torus $W=R_{K/F}^{(1)}\mathbb{G}_m$ with $K/F$ abelian, we compute the $3$-th unramified cohomology $\bar{H}^3_{nr}(F(W),\mathbb Q/\mathbb Z(2))$.
We produce a partial compactification of the variety given by P(t)=N_{K/k}(\mathbf z) whose Brauer group coincides with the unramified Brauer group, where K is an étale k-algebra and P(t)\in k[t] is a nonconstant polynomial. Then we obtain a systematic method to compute the unramified Brauer group for all such varieties.
In this paper, we will calculate the number of Galois extensions of local fields with Galois group A_n and S_n.
We revisit the abstract framework underlying the fibration method for producing rational points on the total space of fibrations over the projective line. By fine-tuning its dependence on external arithmetic conjectures, we render the method unconditional when the degree of the non-split locus is $\leq 2$, as well as in various instances where it is $3$. We are also able to obtain improved results in the regime that is conditionally accessible under Schinzel's hypothesis, by incorporating into it, for the first time, a technique due to Harari for controlling the Brauer--Manin obstruction in families.
We study weak approximation on rationally connected varieties under an assumption of strong approximation for a "simple" variety or under Schinzel's hypothesis. We also get some unconditional results.
We study strong approximation of the equation N_{L/k}(x) = \prod_{i=1}^n p_i(t) where L/k is a finite extension of number fields and p_i(t)'s are distinct irreducible polynomials over k. We prove this equation satisfies strong approximation with Brauer-Manin obstruction when L can be imbedded in k[t]/(p_i(t)) over k for all 1\leq i\leq n. Under Schinzel's hypothesis, we prove that the same result is true without assuming that L can be imbedded in k[t]/(p_i(t)) for all 1\leq i\leq n when L/k is cyclic.
Let X be a toric variety over a number field k with \kbar[X]^\times=\kbar^\times. Let W\subset X be a closed subset of codimension at least 2. We prove that X\setminus W satisfies strong approximation with algebraic Brauer--Manin obstruction.
Ghosh and Sarnak have studied integral points on surfaces defined by an equation x^2+y^2+z^2-xyz= m over the integers. For these affine surfaces, we systematically study the Brauer group and the Brauer-Manin obstruction to the integral Hasse principle. We prove that strong approximation for integral points on any such surface, away from any finite set of places, fails, and that, for m\neq 0, 4, the Brauer group does not control strong approximation.
We give a new version of the open descent theory of Harari and Skorobogatov. As an application of the new version, we prove that some algebraic varieties satisfy strong approximation.
We introduce descent methods to the study of strong approximation on algebraic varieties. We apply them to two classes of varieties defined by P(t)=N_{K/k}(z): firstly for quartic extensions of number fields K/k and quadratic polynomials P(t) in one variable, and secondly for k=Q, an arbitrary number field K and P(t) a product of linear polynomials over Q in at least two variables. Finally, we illustrate that a certain unboundedness condition at archimedean places is necessary for strong approximation.
Let $K/k$ be an extension of number fields, and let $P(t)$ be a quadratic polynomial over $k$. Let $X$ be the affine variety defined by $P(t) = N_{K/k}(\mathbf{z})$. We study the Hasse principle and weak approximation for $X$ in three cases. For $[K:k]=4$ and $P(t)$ irreducible over $k$ and split in $K$, we prove the Hasse principle and weak approximation. For $k=\mathbb{Q}$ with arbitrary $K$, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For $[K:k]=4$ and $P(t)$ irreducible over $k$, we determine the Brauer group of smooth proper models of $X$. In a case where it is non-trivial, we exhibit a counterexample to weak approximation.
For varieties given by an equation N_{K/k}(Ξ)=P(t), where N_{K/k} is the norm form attached to a field extension K/k and P(t) in k[t] is a polynomial, three topics have been investigated: (1) computation of the unramified Brauer group of such varieties over arbitrary fields; (2) rational points and Brauer-Manin obstruction over number fields (under Schinzel's hypothesis); (3) zero-cycles and Brauer-Manin obstruction over number fields. In this paper, we produce new results in each of three directions. We obtain quite general results under the assumption that K/k is abelian (as opposed to cyclic in earlier investigation).
Let k be a number field and K/k Galois. We transform the construction of the unramified Brauer group of the norm one torus R^1_K/k(G_m) into the construction of a special abelian extension over K. If k=Q and K/Q biquadratic, we explicitly construct the unramified Brauer group of R^1_K/Q(G_m).
In this note, we are interested in local-global principles for multinorm equations of the form $\prod_{i=1}^n N_{L_i /k}(z_i) = a$ where $k$ is a global field, $L_i/k$ are finite separable field extensions and $a \in k^*$. In particular, we prove a result relating weak approximation for this equation to weak approximation for some classical norm equation $N_{F/k}(w) = a$ where $F := \bigcap_{i=1}^n L_i$. It provides a proof of a "weak approximation" analogue of a recent conjecture by Pollio and Rapinchuk about multinorm principle. We also provide a counterexample to the original conjecture concerning Hasse principle.
In this note, we give a necessary and sufficient condition for determining which integers can be written as a sum of two integral squares for certain quadratic fields by using the integral Brauer-manin obstruction (see \cite{CTX}). The condition is computable and originally from the reciprocity law.
We construct a finite subgroup of Brauer-Manin obstruction for detecting the existence of integral points on integral models of principle homogeneous spaces of multi-norm tori. Several explicit examples are provided.