arXiv subjects
Publications and source records attributed to Jiangwei Xue.
In this article we construct for each integer $n\ge 2$ an abelian variety $A$ of dimension $n$ defined over a number field for which there exists a symmetric slope sequence of length $2n$ that does not appear as the slope sequence of $\widetilde A$ for all good reductions $\widetilde A$ of $A$.
Let $η: C_{f,N}\to \mathbb{P}^1$ be a cyclic cover of $\mathbb{P}^1$ of degree $N$ which is totally and tamely ramified for all the ramification points. We determine the group of fixed points of the cyclic group $\mathbf{mu}_N\cong \mathbb{Z}/N\mathbb{Z}$ acting on the Jacobian $J_N:=\Jac(C_{f,N})$. For each $\ell$ distinct from the characteristic of the base field, the Tate module $T_\ell J_N$ is shown to be a free module over the ring $\mathbb{Z}_\ell[T]/(\sum_{i=0}^{N-1}T^i)$. We also calculate the degree of the induced polarization on the new part $J_N^{new}$ of the Jacobian.
We classify the endomorphism algebras of factors of the Jacobian of certain hypergeometric curves over a field of characteristic zero. Other than a few exceptional cases, the endomorphism algebras turn out to be either a cyclotomic field $E=\mathbb{Q}(ζ_q)$, or a quadratic extension of $E$, or $E\oplus E$. This result may be viewed as a generalization of the well known results of the classification of endomorphism algebras of elliptic curves over $\mathbb{C}$.
Let K be a field of characteristic zero, f(x) be a polynomial with coefficients in K and without multiple roots. We consider the superelliptic curve C_{f,q} defined by y^q=f(x), where q=p^r is a power of a prime p. We determine the Hodge group of the simple factors of the Jacobian of C_{f,q}.
Let $p$ be a prime, and $q$ a power of $p$. Using Galois theory, we show that over a field $K$ of characteristic zero, the endomorphism algebras of the jacobians of certain superelliptic curves $y^q=f(x)$ are products of cyclotomic fields.
We give lower bounds for dimensions of the centers of Hodge groups of superelliptic jacobians. In "generic case" this bound is precise. We also discuss dimensions of simple abelian subvarieties of superelliptic jacobians.
Suppose that $K$ is a field of characteristic 0, $p$ is an odd prime, $r$ a positive integer, $q=p^r$ a prime power. Suppose that $f(x)$ is a polynomial of degree $n > 4$ with coefficients in $K$ and without multiple roots. Let us consider the superelliptic curve $C: y^q=f(x)$ and its jacobian $J(C)$. Assuming that $K$ is a subfield of the field of complex numbers, we study the (connected reductive algebraic) Hodge group $Hdg$ of the corresponding complex abelian variety $J(C)$. In our previous paper (arXiv:0907.1563 [math.AG]) we studied the center of $Hdg. In this paper we study the semisimple part (commutator subgroup) of $Hdg$. Assuming that $p$ does not divide $n$ and $n-1$ is not divisible by $q$, the Galois group of $f(x)$ over $K$ is either the full symmetric group $S_n$ or the alternating group $A_n$, we prove that the semisimple part of $Hdg$ is "as large as possible".