Endomorphism algebras of Jacobians of certain superelliptic curves
Jiangwei Xue
设$p$为素数，$q$为$p$的幂。利用伽罗瓦论，证明了在特征为零的域$K$上，某些超椭圆曲线$y^q=f(x)$的Jacobian的自同态代数是分圆域的积.
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.
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Centers of Hodge groups of superelliptic jacobians
给出了超椭圆雅可比算子的Hodge群的中心维数的下界。在“一般情况”下，这个界限是精确的。我们还讨论了超椭圆雅可比算子的单交换子簇的维数。
Superelliptic jacobians
讨论了超椭圆曲线y^q=f（x）的Jacobian的自同态代数的结构，其中q是特征为零的有理函数k（t）域上的一个三次或四次不可约多项式，f（x）的伽罗瓦群是全对称群.
We give a structure theorem for the $m$-torsion of the Jacobian of a general superelliptic curve $y^m=F(x)$. We study existence of torsion on curves of the form $y^q=x^p-x+a$ over finite fields of characteristic $p$. We apply those results to bound from below the Mordell-Weil ranks of Jacobians of certain superelliptic curves over $\mathbb Q$.
Let K be a field of characteristic zero, n>4 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group is doubly transitive simple non-abelian group. Let p be an odd prime, Z[ζ_p] the ring of integers in the p-th cyclotomic field, C_{f,p}:y^p=f(x) the corresponding superelliptic curve and J(C_{f,p}) its jacobian. Assuming that either n=p+1 or p does not divide n(n-1), we prove that the ring of all endomorphisms of J(C_{f,p}) coincides with Z[ζ_p].
In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic curves of fixed genus $g\geq 8$ with CM Jacobians. The proof relies on the geometric structures of Shimura subvarieties in Siegel modular varieties and the stability properties of Higgs bundles associated to fibred surfaces.
Non-isogenous superelliptic jacobians
在他以前的文件（J。Math.544（2002），91- 110; Math.AG/0103203）给出了超椭圆Jacobian的一种显式构造，其自同态环是$p$第分圆域中的整数环. (Here $p$是一个奇素数。）在本文中，我们讨论当这些雅可比矩阵是相互非同构的。(The在作者的电子印刷数学中处理了超椭圆Jacobian的情况。NT/0301173。
Given a polynomial $f\in\mathbb{C}[x]$, we consider the family of superelliptic curves $y^d=f(x)$ and their Jacobians $J_d$ for varying integers $d$. We show that for any integer $g$ the number of abelian varieties up to isogeny of dimension $\le g$ which appear in any $J_d$ is finite and their multiplicities are bounded.
Hodge Groups of certain Superelliptic Jacobians II
设K是特征为零的域，f（x）是系数在K中且无重根的多项式.本文考虑由y^q=f（x）定义的超椭圆曲线C_{f,q}，其中q=p^r是素数p的幂，确定了C_{f,q}的雅可比矩阵的单因子的Hodge群.
This is (mostly) a survey article. We use an information about Galois properties of points of small order on an abelian variety in order to describe its endomorphism algebra over an algebraic closure of the ground field. We discuss in detail applications to jacobians of cyclic covers of the projective line.
Hodge groups of certain superelliptic jacobians
设$K$是特征为0的域，$p$是奇素数，$r$是正整数，$q=p^r$是素数幂。假设$f(x)$是一个次数为$n > 4$的多项式，系数在$K$中，没有重根。让我们考虑超椭圆曲线$C: y^q=f(x)$和它的雅可比$J(C)$。假设$K$是复数域的一个子域，我们研究了相应的复交换簇$J(C)$的（连通约化代数）Hodge群$Hdg$。在我们以前的论文（arXiv：0907.1563 [math.AG]）中，我们研究了$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$是“尽可能大”的中心.
On Jacobians of curves with superelliptic components
研究了超椭圆曲线的Jacobian的自同构分解。对于方程为$y^n=f(x^m)$的曲线，给出了曲线自同构诱导的雅可比矩阵分解的$m$和$n$形式的充要条件.此外，我们推广了一族非超椭圆曲线$\mathcal X_{r,s} $在\cite{Ya}中的构造，并确定了Jacobian $\mbox{Jac} (\mathcal X_{r, s})$分解$r$和$s$的算术条件。
Let $K$ be a field of prime characteristic $p$, $n>4 $ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Let $l$ be an odd prime different from $p$, $Z[ζ_l]$ the ring of integers in the $l$th cyclotomic field, $C_{f,l}:y^l=f(x)$ the corresponding superelliptic curve and $J(C_{f,l})$ its jacobian. We prove that the ring of all endomorphisms of $J(C_{f,l})$ coincides with $Z[ζ_l]$ if $J(C_{f,l})$ is an ordinary abelian variety and $(l,n)\ne (5,5)$.
Superelliptic jacobians and central simple representations
本文在多项式不可约、Galois 群双传递且极大子群指数满足条件时，证明超椭圆曲线 y^p=f(x) 的 Jacobian 自同态环为第 p 个分圆环。
In 2016, Yuri Zarhin gave formulas for "dividing a point on a hyperelliptic curve by 2." Given a point $P$ on a hyperelliptic curve $\mathcal{C}$, Zarhin gives the Mumford's representation of every degree $g$ divisor $D$ such that $2(D - g \infty) \sim P - \infty$. The aim of this paper is to generalize Zarhin's result to the superelliptic situation; instead of dividing by 2, we divide by $1 - ζ$. Even though there is no Mumford's representation for superelliptic curves, we give a formula for functions which cut out $D$.
From hyperelliptic to superelliptic curves
本文综述了将椭圆与超椭圆曲线理论推广到所有超椭圆曲线的框架，涵盖自同构群、模空间分层、不变量、最小模型、雅可比簇等核心内容，并总结了最新进展与开放问题。
The "exponential" torsion of superelliptic Jacobians
本文研究超椭圆曲线雅可比簇的$\ell$-幂次挠子群，在温和条件下确定相关$\ell$-进表示的像，证明其行列式像含于有限指数显式$\n\mathbb Z_{\ell}$-格，并验证了一般超椭圆雅可比簇的Hodge、Tate与Mumford-Tate猜想。
Fixed points and homology of superelliptic Jacobians
设$η: C_{f,N}\to \mathbb{P}^1$是$N$次的$\mathbb{P}^1$的循环覆盖，它对所有分歧点都是完全和光滑分歧的.我们确定了循环群$\mathbf{mu}_N\cong \mathbb{Z}/N\mathbb{Z}$作用于雅可比矩阵$J_N:=\Jac(C_{f,N})$的不动点群。对于每个$\ell$不同于基本字段的特性，泰特模块$T_\ell J_N$被示出为是环$\mathbb{Z}_\ell[T]/(\sum_{i=0}^{N-1}T^i)$上的自由模块。我们还计算了雅可比矩阵的新部分$J_N^{new}$上的激发极化度。
Non-isogenous superelliptic jacobians II
本文研究超椭圆曲线的雅可比簇等源性问题，证明当两个多项式一个不可约一个可约时，若对应雅可比簇在代数闭包上等源，则两个自同态代数均含乘法阶为n的可逆元素。
Superelliptic curves with many automorphisms and CM Jacobians
本文完整分类了具有多自同构的超椭圆曲线，并逐一判定其雅可比簇是否具有复乘，进而证明了Streit复乘准则的逆命题。
We describe several improvements to algorithms for the rigorous computation of the endomorphism ring of the Jacobian of a curve defined over a number field.
Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/\mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian of $C$. By work of Pries--Ulmer, $J$ satisfies the Birch and Swinnerton-Dyer conjecture (BSD). Generalizing work of Griffon--Ulmer, we compute the $L$-function of $J$ in terms of certain Gauss sums. In addition, we estimate several arithmetic invariants of $J$ appearing in BSD, including the rank of the Mordell--Weil group $J(\mathbb F_{r}(t))$, the Faltings …
Endomorphism algebras of hyperelliptic jacobians and finite projective lines
证明了超椭圆曲线y^2=f（x）的雅可比行列式是绝对单的，如果deg（f）=q+1，其中q是5模8的幂素同余，多项式f（x）在特征为零的基域上不可约，其伽罗瓦群是L_2（q）.
Let $C$ be a curve of genus at least three defined over a number field, and let $r$ be the rank of the rational points of its Jacobian. Under mild hypotheses on $r$, recent results by Katz, Rabinoff, Zureick-Brown, and Stoll bound the number of rational points on $C$ by a constant that depends only on its genus. Yet one expects an even stronger bound that depends favorably on $r$: when $r$ is small, there should be fewer points on $C$. In a 2013 paper, Stoll established such a "rank-favorable" bound for hyperelliptic curves using Chabauty's method. In the present work we extend Stoll's results…
Cyclic covers of prime power degree, jacobians and endomorphisms
设$K$是特征为0的域，$K_a$是它的代数闭包，$p$是素数，$q=p^r$是幂素数.假设$f(x) \in K[x]$是一个没有重根的$n > 4$次多项式。让我们考虑超椭圆曲线$C: y^q=f(x)$及其雅可比矩阵$J(C)$。研究了$J(C)$的所有$K_a$ -自同态的自同态代数$End^0(J(C))$.证明了如果$f$在$K$上的Galois群是全对称群$S_n$或交错群$A_n$，则$End^0(J(C))$是“尽可能小的”.
This paper is the first version of a project of classifying all superelliptic curves of genus $g \leq 48$ according to their automorphism group. We determine the parametric equations in each family, the corresponding signature of the group, the dimension of the family, and the inclussion among such families. At a later stage it will be determined the decomposition of the Jacobians and each locus in the moduli spaces of curves.
We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute $\operatorname{End}_{\overline{K}}(A)$ when $A$ is the Jacobian of a nice genus-2 curve over a number field $K$. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of $\operatorname{Jac}(C)$ given in the LMFDB ($L$-functions and modular forms database) is correct for all the genus 2 curves $C$ currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases.
Pairings on Jacobians of Hyperelliptic Curves
考虑定义在有限域上的超椭圆亏格2曲线的雅可比矩阵。在雅可比矩阵的自同态环的一定限制条件下，给出了雅可比矩阵上所有非退化、双线性、反对称和伽罗瓦不变对的一个显式刻画。从这一描述可以得出，没有这样的配对可以比Weil配对更有效地计算。为了建立这个结果，我们需要一个明确的描述的表示Frobenius自同态的l-挠子群的雅可比。给出了该描述。特别是，我们表明，如果特征多项式的Frobenius自同态分裂成线性因子模l，那么Frobenius是diagonalizable。最后，在Frobenius元是自同态环的某个子环的元素的限制下，证明了如果Frobenius自同态的特征多项式分裂为模l的线性因子，则Jacobian关于l的嵌入度与总嵌入度是相同的.
The endomorphism rings of jacobians of cyclic covers of the projective line
设K是特征为0的域，$K_a$是它的代数闭包，p是奇素数.假设$f(x) \in K[x]$是一个没有重根的$n \ge 5$次多项式。让我们考虑曲线$C: y^p=f(x)$及其雅可比矩阵J（C）。已知J（C）的所有$K_a$ -自同态的环End（J（C））包含第p个分圆域（由C的明显自同构生成）中的整数环$Z[ζ_p]$。本文证明了如果K上f的伽罗瓦群是对称群$S_n$或交错群$A_n$，则有$End(J(C))=Z[ζ_p]$.
Cyclic covers of the projective line, their jacobians and endomorphisms
研究曲线$y^p=f(x)$的复雅可比矩阵$J(C)$的自同态环$End(J(C))$，其中$p$是奇素数，$f(x)$是一个无重根的复系数$n>4$次多项式.假设$f$的所有系数都位于一个（子）域$K$中，并且$K$上的$f$的伽罗瓦群是全对称群$S_n$或交替群$A_n$。然后证明了$End(J(C))$是在$p$ th分圆域上的整数环，如果$p$是Fermat素数（例如，$p=3,5,17,257$）。作者在Math. Res. Lett. 7（2000），123- 132.
Counting points on superelliptic curves in average polynomial time
本研究实现了一种计算有理数域上超椭圆曲线上点数的平均多项式时间算法，通过计算Cartier–Manin矩阵的迹提升效率，还可得到该矩阵以确定雅可比簇的p-秩及模pζ函数分子。
In [5], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively in [1] to study the algebraic curves for which their Jacobian has superelliptic components. In this note, we determine the correct version of the genus formula with an algebraic proof. Then, we show that the formula given in [5] works only under certain conditions.
Endomorphisms of polynomial rings and Jacobians
由于第10页的一个关键论点错误，作者撤回了这篇论文。
Hyperelliptic jacobians without complex multiplication
证明了当多项式f的伽罗瓦群"非常大“时，超椭圆曲线y^2=f（x）的雅可比行列式在特征为零的基域的代数闭包上没有非平凡的自同态.
We give a quasihomogeneity criterion for Gorenstein curves. For complete intersections, it is related to the first step of Vasconcelos' normalization algorithm. In the process, we give a simplified proof of the Kunz-Ruppert criterion.
Given a polynomial $W$ with an isolated singularity, we can consider the Jacobian ring as an invariant of the singularity. If in addition we have a group action on the polynomial ring with $W$ fixed, we are led to consider the twisted Jacobian ring which reflects the equivariant structure as well. Our main result is to show that the twisted Jacobian ring is isomorphic to an endomorphism ring of the "twisted diagonal" matrix factorization. As an application, we suggest a way to investigate Floer theory of Lagrangian submanifolds which represent homological mirror functors.
Explicit endomorphism of the Jacobian of a hyperelliptic function field of genus 2 using base field operations
我们提出了一个有效的自同态的雅可比曲线$C$的亏格2（超椭圆）的因子有一个不相交的支持。这扩展了Costello和Lauter在[12]中的工作，他们在$\mathbb{J}(C)$中仅使用基本字段操作计算了具有不相交支持的除数加倍和除数相加的显式公式。对于第三种情况，给出了明确的公式，对于除数加倍，给出了稍微不同的方法。
In this work we consider constructions of genus three curves $X$ such that $\mathrm{End}(\mathrm{Jac} (X))\otimes Q$ contains the totally real cubic number field $Q(ζ_7 +\barζ_7 )$. We construct explicit three-dimensional families whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when $X$ is hyperelliptic was studied in a previous work by Hoffman and Wang and some nonhyperelliptic curves were constructed in a previous paper by Hoffman, Z. Liang. Sakai and Wang.
In this work we consider constructions of genus three curves $X$ such that $\mathrm{End}(\mathrm{Jac}(X)) \otimes Q$ contains the totally real cubic number field $Q(ζ_ 7 + \overlineζ_7)$. We construct explicit two-dimensional families defined over $Q(s; t)$ whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when X is hyperelliptic was studied by the authors Hoffman and Wang in a previous work. We calculate the zeta function of one of these curves. Conjecturally this zeta function is described by a modular form.
Let $G$ be a finite group acting effectively on the complex affine plane. If the $G$-action commutes with an étale endomorphism $f$ of the affine plane and the order of $G$ is even then the endomorphism $f$ is an automorphism.
Some remarks on the Jacobian conjecture and polynomial endomorphisms
本文首先证明了齐次Keller映射在通过原点的直线上是内射的。我们随后推导出一个推广，即在某些条件下，具有正次数的$r$齐次部分的多项式自同态不具有$r$乘以通过原点的直线上的同一像点，以防其雅可比行列式在该直线上的任何地方都不消失。因此，如果$r$是基域中的一个单位，则次数为$r$的凯勒映射在$r > 1$共线点上不取相同的值。其次，我们证明了对于可逆映射$x + H$的度$d$，使得$\ker \jac H$在基域上具有$n-r$独立向量，特别是对于可逆幂线性映射$x + (Ax)^{*d}$与$\rk A = r$，$x + H$的逆的度至多为$d^r$。
Hyperelliptic jacobians without complex multiplication in positive characteristic
证明了当偶次多项式$f$的伽罗瓦群"非常大”时，超椭圆曲线$y^2=f(x)$的Jacobian在奇特征上在基域的代数闭包上没有非平凡的自同态.特征零的情况以前由作者处理过（Math. Res. Letters 7（2000），123- 132）。
Families of absolutely simple hyperelliptic jacobians
证明了超椭圆曲线$y^2=(x-t)h(x)$的雅可比行列式在特征为零的基域$K$的代数闭包上没有非平凡的自同态，如果$t \in K$和$K$上多项式$h(x)$的伽罗瓦群“非常大”，且$deg(h)$是大于8的偶数. (The根据作者以前的结果，可以很容易地得出奇数$deg(h)>3$的情况。）
We give practical numerical methods to compute the period matrix of a plane algebraic curve (not necessarily smooth). We show how automorphisms and isomorphisms of such curves, as well as the decomposition of their Jacobians up to isogeny, can be calculated heuristically. Particular applications include the determination of (generically) non-Galois morphisms between curves and the identification of Prym varieties.
Ranks of Jacobians in towers of function fields
设$k$是特征为零的域，$K=k(t)$是$k$上的有理函数域.本文结合Ulmer关于$K$上某些Jacobian秩的公式和Zarhin关于Jacobian自同态的强上界，给出了$k(t)$上高维、绝对单Jacobian在塔$k(t^{1/p^r})$中具有有界秩的许多例子.在许多情况下，我们能够计算塔的每一层的等级。
Let $\ell$ be a rational prime and $k$ a number field. Given a superelliptic curve $C/k$ of $\ell$-power degree, we describe the field generated by the $\ell$-power torsion of the Jacobian variety in terms of the branch set and reduction type of $C$ (and hence, in terms of data determined by a suitable affine model of $C$). If the Jacobian is good away from $\ell$ and the branch set is defined over a pro-$\ell$ extension of $k(μ_{\ell^\infty})$ unramified away from $\ell$, then the $\ell$-power torsion of the Jacobian is rational over the maximal such extension. By decomposing the covering int…
Efficiently computable endomorphisms for hyperelliptic curves
椭圆曲线有一个众所周知的和明确的理论，用于构造和应用自同态，可以应用于提高性能的标量乘法。最近的工作已经扩展这些技术的超椭圆雅可比，但一个障碍是缺乏明确的曲线模型，以及一个有效的可计算的自同态。在超椭圆曲线的情况下，有有限的例子，大多数方法集中在特殊的CM曲线或在一个小的字段上定义的曲线。在这篇文章中，我们描述了三个无限族的曲线，承认一个有效的可计算的自同态，并给出算法，其有效的应用。
On endomorphisms of affine spaces and the Jacobian problem
本文研究仿射空间平展自同态的纤维基数、像补集及几何次数，构造反例否证若干猜想，并证明雅可比猜想在特定复合条件下成立。
We give uniformizations of the Klein quartic curve and the Fermat septic curve as Shimura curves parametrizing Abelian $6$-folds with endomorphisms $\ZZ[\z_7]$.
A curve over a field of characteristic $p$ is called ordinary if the $p$-torsion of its Jacobian as large as possible, that is, an $\mathbb{F}_p$ vector space of dimension equal to its genus. In this paper we consider the following question: fix a finite field $\mathbb{F}_q$ and a family $\mathscr{F}$ of curves over $\mathbb{F}_q$. Then, what is the probability that a curve in this family is ordinary? We answer this question when $\mathscr{F}$ is either the Artin-Schreier family in any characteristic or a superelliptic family in characteristic 2.
Non-supersingular Hyperelliptic jacobians
在以前的文章中，作者证明了超椭圆曲线C：y^2=f（x）的Jacobian J（C）在基域K的代数闭包K_a上的特征不等于2，只有平凡的自同态，如果f（x）在K[x]中的不可约多项式的伽罗瓦群是全对称群S_n或交错群A_n.这里n > 8是f的次数。本文的目的是将这一结果推广到n= 7，8或n= 5，6且char（K）>3的情形。
Hyperelliptic jacobians and projective linear Galois groups
在他以前的论文（Math.Res.Letters 7（2000），123 - 132）中，作者证明了，在特征零处，超椭圆曲线$C: y^2=f(x)$的雅可比$J(C)$在基域$K$的代数闭包$K_a$上只有平凡的自同态，如果不可约多项式$f(x) \in K[x]$的伽罗瓦群$Gal(f)$是对称群$S_n$或交替群$A_n$。这里$n>4$是$f$的度数。在数学AG/0003002中，我们把这个结果推广到某些"较小"的伽罗瓦群的情况。特别地，我们处理了无穷级数$n=2^r+1, Gal(f)=L_2(2^r)$和$n=2^{4r+2}+1, Gal(f)=Sz(2^{2r+1})$。本文证明了$J(C)$在$K_a$上只有平凡自同态，如果$f$的根的集合可以与具有奇特征的有限域$F_q$上的$(m-1)$维射影空间$P^{m-1}(F_q)$相一致，使得作为其置换群的$Gal(f)$成为射影线性群$PGL(m,F_q)$或射影特殊线性群$L_m(q):=PSL(m,F_q)$.在这里，我们假设$m>2$。
We study the Selmer varieties of smooth projective curves of genus at least two defined over $\mathbb{Q}$ which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and Kim to show that Kim's non-abelian Chabauty method applies to such a curve. By combining this with results of Bogomolov-Tschinkel and Poonen on unramified correspondences, we deduce that any cover of $\mathbf{P}^1$ with solvable Galois group, and in particular any superelliptic curve over $\mathbb{Q}$, has only finitely many rational points over $\mathbb{Q}$.
Jedrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conj…
Endomorphisms preserving coordinates of polynomial algebras
证明了k[x_1，…，x_n]在特征为零的域k上取每个tame坐标到一个坐标，必须是k中的一个非零常数。本文还证明了A：=R[x_1，…，x_n]（其中R是无限域k上的有限变元数的多项式环）是k中的一个非零常数。
A polynomial endomorphism $σ\in {\rm End}_K(P_n)$ is called a Jacobian map if its Jacobian is a nonzero scalar (the field has zero characteristic). Each Jacobian map $σ$ is extended to an endomorphism $σ$ of the Weyl algebra $A_n$. The Jacobian Conjecture (JC) says that every Jacobian map is an automorphism. Clearly, the Jacobian Conjecture is true iff the twisted (by $σ$) $P_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. It is shown that the $A_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. Furthermore, the $A_n$-module ${}^σ P_n$ is holonomic and as a result h…
On abelian surfaces with potential quaternionic multiplication
域K上的阿贝尔曲面A有势四元数乘法，如果A的几何自同态环End_ \bar K（A）是不定有理除四元数代数中的一个阶.在这个简短的说明中，我们研究了这些曲面的自同态环的可能结构，我们提供了明确的例子，雅可比曲线的亏格2，这表明我们的结果是尖锐的。
Hyperelliptic jacobians and $\U_3(2^m)$
在他以前的论文（Math. Res. Letters 7（2000），123- 132）中，作者证明了，在特征零处，超椭圆曲线$C: y^2=f(x)$的雅可比$J(C)$在基域$K$的代数闭包$K_a$上只有平凡的自同态，如果不可约多项式$f(x) \in K[x]$的伽罗瓦群$Gal(f)$是对称群$S_n$或交替群$A_n$。这里$n>4$是$f$的度数。在数学AG/0003002中，我们将这个结果推广到某些“较小”伽罗瓦群的情况。特别地，我们处理了无穷级数$n=2^r+1, Gal(f)=L_2(2^r)$和$n=2^{4r+2}+1, Gal(f)=Sz(2^{2r+1})$。在本文中，我们做的情况下，$Gal(f)=\U_3(2^m)$和$n=2^{3m}+1$。
A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its étale endomorphisms are proper. We study the conjecture for $\mathbb{Q}$-acyclic surfaces of negative Kodaira dimension. We show that $G$-equivariant counterexamples for infinite group $G$ exist if and only if $G=\mathbb{C}^*$ and we classify them relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected $\mathbb{C}^*$-surfaces of negative Kodaira dimension which admit non-proper $\mathbb{C}^*$-equivariant étale endomorphisms. We prove also that for every integers $r\geq …
Monodromy groups of Jacobians with definite quaternionic multiplication
本文利用带定四元数乘法的超椭圆 Jacobi 簇显式族，证明对任意偶数 $g\ge4$，$\mathbb{Q}$ 上存在无穷多个几何不同源的 $g$ 维阿贝尔簇，其连通单值域严格大于自同态定义域。
The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture
关于$n$维的雅可比猜想断言，在特征为零的域上，$n$维仿射空间的任何多项式自同态，当雅可比等于1时，都是可逆的。秩为$n$的Dixandian猜想断言$n$阶Weyl代数（$n$变量中的多项式微分算子代数）的任何自同态都是可逆的。证明了维度$2n$上的Jacobian猜想蕴含秩$n$上的Dixmier猜想。再加上一个著名的相反方向的含义，它表明稳定的雅可比矩阵和Dixobian矩阵是等价的。证明的主要工具是有限特征化约。在论文完成后，我们了解到主要结果已经由Y.Tsuchimoto在Osaka Journal of Mathematics第42卷第2期（2005年6月）上发表。他的证明是不同的。
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