arXiv subjects
Publications and source records attributed to Dasheng Wei.
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.
The asymptotic formula of the number of integral points in non-compact symmetric homogeneous spaces of semi-simple simply connected algebraic groups is given by the average of the product of the number of local solutions twisted by the Brauer-Manin obstruction. The similar result is also true for homogeneous spaces of reductive groups with some restriction. As application, we will give the explicit asymptotic formulae of the number of integral points of certain norm equations and explain that the asymptotic formula of the number of integral points in Theorem 1.1 of \cite{EMS} is equal to the product of local integral solutions over all primes and answer a question raised by Borovoi related to the example 6.3 in \cite{BR95}.
We construct a finite subgroup of Brauer-Manin obstruction for detecting the existence of integral points on integral models of homogeneous spaces of linear algebraic groups of multiplicative type. As application, the strong approximation theorem for linear algebraic groups of multiplicative type is established. Moreover, the sum of two integral squares over some quadratic fields is discussed.
We propose a method to determine the solvability of the diophantine equation $x^2-Dy^2=n$ for the following two cases: $(1)$ $D=pq$, where $p,q\equiv 1 \mod 4$ are distinct primes with $(\frac{q}{p})=1$ and $(\frac{p}{q})_4(\frac{q}{p})_4=-1$. $(2)$ $D=2p_1p_2... p_m$, where $p_i\equiv 1 \mod 8,1\leq i\leq m$ are distinct primes and $D=r^2+s^2$ with $r,s \equiv \pm 3 \mod 8$.
We propose a method for determining which integers can be written as a sum of two integral squares for quadratic fields $\Q(\sqrt{\pm p})$, where $p$ is a prime.
Results on $L$-functions of Artin-Schreier coverings by Dwork, Bombieri and Adolphson-Sperber are generalized to $L$-functions of Witt coverings.