Riemann surfaces with a large abelian group of automorphisms

Primitive automorphisms of a simple abelian variety

We shall prove that an automorphism of a simple abelian variety is primitive if and only if it is of infinite order.

固定维数的复单阿贝尔簇自同构的第一动力次数的最小值

The minimum value of the first dynamical degrees of automorphisms of complex simple abelian varieties with fixing dimensions

该研究针对复单阿贝尔簇自同构的第一动力次数，分别探讨了素数维情形下大于1的最小值，以及维数为2到10时大于1的最小值的计算问题。

Prime order automorphisms of abelian surfaces: a lattice-theoretic point of view

In this article, we focus on a new perspective of automorphisms of complex 2-tori, reviewing previous works from a lattice-theoretic point of view. In particular, we give a classification of families of symplectic and non-symplectic automorphisms of prime order.

Lifting Automorphisms on Abelian Varieties as Derived Autoequivalences

We show that on an Abelian variety over an algebraically closed field of positive characteristic, the obstruction to lifting an automorphism to an Abelian variety over a field of characteristic zero as a morphism vanishes if and only if it vanishes for lifting it as an autoequivalence. We also compare the deformation space of these two types of deformations.

On Riemann surfaces of genus $g$ with $4g-4$ automorphisms

In this article we study compact Riemann surfaces with a non-large group of automorphisms of maximal order; namely, compact Riemann surfaces of genus $g$ with a group of automorphisms of order $4g-4.$ Under the assumption that $g-1$ is prime, we provide a complete classification of them and determine isogeny decompositions of the corresponding Jacobian varieties.

复简单阿贝尔簇自同构的动力次数与Salem数

Dynamical degrees of automorphisms of complex simple abelian varieties and Salem numbers

本文证明每个Salem数可实现为复简单阿贝尔簇自同构的第一动力次数，并证明固定维数时该次数集合除1外有最小值，且存在自同构使其第一动力次数尽可能接近1。

Abelian surfaces with an automorphism and quaternionic multiplication

We construct one dimensional families of Abelian surfaces with quaternionic multiplication which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)-polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space which parametrizes Abelian surfaces with real multiplication by Z[\sqrt{2}].

Riemann surfaces of genus $1+q^2$ with $3q^2$ automorphisms

In this article we classify compact Riemann surfaces of genus $1+q^2$ with a group of automorphisms of order $3q^2,$ where $q$ is a prime number. We also study decompositions of the corresponding Jacobian varieties.

On Riemann Surfaces of genus g with 4g automorphisms

We determine, for all genus $g\geq2$ the Riemann surfaces of genus $g$ with $4g$ automorphisms. For $g\neq$ $3,6,12,15$ or $30$, this surfaces form a real Riemann surface $\mathcal{F}_{g}$ in the moduli space $\mathcal{M}_{g}$: the Riemann sphere with three punctures. The set of real Riemann surfaces in $\mathcal{F}_{g}$ consists of three intervals its closure in the Deligne-Mumford compactification of $\mathcal{M}_{g}$ is a closed Jordan curve.

A note on large automorphism groups of compact Riemann surfaces

Belolipetsky and Jones classified those compact Riemann surfaces of genus $g$ admitting a large group of automorphisms of order $λ(g-1)$, for each $λ>6,$ under the assumption that $g-1$ is a prime number. In this article we study the remaining large cases; namely, we classify Riemann surfaces admitting $5(g-1)$ and $6(g-1)$ automorphisms, with $g-1$ a prime number. As a consequence, we obtain the classification of Riemann surfaces admitting a group of automorphisms of order $3(g-1)$, with $g-1$ a prime number. We also provide isogeny decompositions of their Jacobian varieties.

Realisation of abelian varieties as automorphism groups

Let $A/F$ be an abelian variety over a field. Does there exist a smooth projective $F$-variety $X$, such that $A$ is isomorphic to the automorphism group scheme of $X/F$? We show that the answer is positive, if and only if $A$ has only finitely many automorphisms, over an algebraic closure of $F$. When $F=\mathbb C$, this result is due to Lombardo and Maffei. When $F$ is algebraically closed, it was obtained independently by Blanc and Brion.

Equations of Riemann surfaces with automorphisms

We present an algorithm for computing equations of canonically embedded Riemann surfaces with automorphisms. A variant of this algorithm with many heuristic improvements is used to produce equations of Riemann surfaces $X$ with large automorphism groups (that is, $|\mathrm{Aut}(X)| > 4(g_X-1)$) for genus $4 \leq g_X \leq 7$. The main tools are the Eichler trace formula for the character of the action of $\mathrm{Aut}(X)$ on holomorphic differentials, algorithms for producing matrix generators of a representation of a finite group with a specified irreducible character, and Gröbner basis techni…

On families of Riemann surfaces with automorphisms

In this article we determine the maximal possible order of the automorphism group of the form $ag + b$, where $a$ and $b$ are integers, of a complex three and four-dimensional family of compact Riemann surfaces of genus $g$, appearing for all genus. In addition, we construct and describe explicit complex three and four-dimensional families possessing these maximal number of automorphisms.

Nilpotent groups of automorphisms of families of Riemann surfaces

In this article we extend results of Zomorrodian to determine upper bounds for the order of a nilpotent group of automorphisms of a complex $d$-dimensional family of compact Riemann surfaces, where $d \geqslant 1.$ We provide conditions under which these bounds are sharp. In addition, for the one-dimensional case we construct and describe an explicit family attaining the bound for infinitely many genera. We obtain similar results for the case of $p$-groups of automorphisms.

Riemann Surfaces of genus g with an automorphism of order p prime and p>g

The present work completes the classification of the compact Riemann surfaces of genus g with an analytic automorphism of order p (prime number) and p > g. More precisely, we construct a parameteriza- tion space for them, we compute their groups of uniformization and we compute their full automorphism groups. Also, we give affine equations for special cases and some implications on the components of the singular locus of the moduli space of smooth curves of genus g.

On the one-dimensional family of Riemann surfaces of genus $q$ with $4q$ automorphisms

Bujalance, Costa and Izquierdo have recently proved that all those Riemann surfaces of genus $g \ge 2$ different from $3, 6, 12, 15$ and 30, with exactly $4g$ automorphisms form an equisymmetric one-dimensional family, denoted by $\mathcal{F}_g.$ In this paper, for every prime number $q \ge 5,$ we explore further properties of each Riemann surface $S$ in $\mathcal{F}_q$ as well as of its Jacobian variety $JS.$

关于一般超奇异阿贝尔簇自同构的Oort猜想

Oort's conjecture on automorphisms of generic supersingular abelian varieties

本文证明了Oort猜想：在特征p的亏格g主极化阿贝尔簇模空间的超奇异轨迹上，除g=2或3且p=2的情况外，通用主极化阿贝尔簇的自同构群仅为±1。

具有 $2p^2$ 个自同构的三角黎曼曲面的分类

A classification of triangular Riemann surfaces with $2p^2$ automorphisms

本文分类并描述了具有 $2p^2$ 阶群三角作用的紧黎曼曲面（$p$ 为奇素数），证明所有此类曲面同构于有理数域上的曲线，并推导出定向保持自同构群阶为 $2p^2$ 的定向正则超地图的分类。

Automorphisms of Abelian Varieties and Principal Polarizations

The Narasimhan-Nori conjecture asks for a closed formula for the number of non-isomorphic principal polarizations of any given abelian variety. In this paper, we introduce a new algorithm that gives a lower bound on the number of non-isomorphic principal polarizations on any given abelian variety. We show, for example, that the Jacobian of the genus four underlying curve of Schoen's I-WP minimal surface has at least 9 non-isomorphic principal polarizations. We also explore the Jacobians of Klein's quartic, Fermat's quartic, Bring's curve, and more.

Two moduli spaces of abelian fourfolds with an automorphism of order five

We find explicit projective models of a compact Shimura curve and of a (non-compact) surface which are the moduli spaces of principally polarised abelian fourfolds with an automorphism of order five. The surface has a 24-nodal canonical model in IP^4 which is the complete intersection of two S_5-invariant cubics. It is dominated by a Hilbert modular surface and we give a modular interpretation for this. We also determine the L-series of these varieties as well as those of several modular covers of the Shimura curve.

Abel 曲面上点的 Hilbert 概形的自同构

Automorphisms of Hilbert Schemes of Points on Abelian Surfaces

本文构造 Abel 曲面任意点数 Hilbert 概形上保持例外除子的非自然自同构，并证明 Picard 秩 1 的主极化或极化自交为平方数时两点情形的所有自同构均自然。

A class of pseudoreal Riemann surfaces with diagonal automorphism group

A Riemann surface $\mathcal{S}$ having field of moduli $\mathbb{R}$, but not a field of definition, is called \emph{pseudoreal}. This means that $\mathcal{S}$ has anticonformal automorphisms, but non of them is an involution. We call a Riemann surface $\mathcal{S}$ \emph{plane} if it can be described by a smooth plane model of some degree $d\geq4$ in $\mathbb{P}^2_{\mathbb{C}}$. We characterize pseudoreal-plane Riemann surfaces $\mathcal{S}$, whose conformal automorphism group $\operatorname{Aut}_+(\mathcal{S})$ is $\operatorname{PGL}_3(\mathbb{C})$-conjugate to a finite non-trivial group $G$ …

广义准二面体群作为黎曼面的自同构群

Generalized quasi-dihedral group as automorphism group of Riemann surfaces

本文研究了广义准二面体群在黎曼面上的共形作用，并探讨了其作用的唯一性问题。

Elliptic K3 surfaces with abelian and dihedral groups of symplectic automorphisms

We analyze K3 surfaces admitting an elliptic fibration $E$ and a finite group $G$ of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration $E/G$ comparing its properties to the ones of $E$. We show that if $E$ admits an $n$-torsion section, its quotient by the group of automorphisms induced by this section admits again an $n$-torsion section. Considering automorphisms coming from the base of the fibration, we can describe the Mordell--Weil lattice of a fibration described by Kloosterman. We give the isometries between lattices described by th…

亏格为 $2(p-1)$ 且具有 $4p$ 个自同构的 Riemann 曲面的轨迹

The locus of Riemann surfaces of genus $2(p-1)$ with $4p$ automorphisms

本文完整显式描述了亏格为 $2(p-1)$ 且具有阶为 $4p$ 自同构群的紧 Riemann 曲面轨迹，给出了相应 Jacobi 簇的同源分解，并探讨了最具对称性曲面的复乘与实乘性质。

On automorphism groups of fiber bundles

We obtain analogues of classical results on automorphism groups of holomorphic fiber bundles, in the setting of group schemes. Also, we establish a lifting property of the connected automorphism group, for torsors under abelian varieties. These results will be applied to the study of homogeneous bundles over abelian varieties.

Automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fields

We prove that the automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fields are cyclic, and give a complete list of finite groups that can be realized as such automorphism groups.

On Abelian Automorphism Groups of Mumford Curves

We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abelian subgroups of the automorphism group of a Mumford curve in terms of its genus.

Automorphisms of ${\mathbb C}^*$ Moduli Spaces Associated to a Riemann Surface

We compute the automorphism groups of the Dolbeault, de Rham and Betti moduli spaces for the multiplicative group ${\mathbb C}^*$ associated to a compact connected Riemann surface.

On Abelian Automorphism Groups of Hypersurfaces

Given integers $d\ge 3$ and $N\ge 3$. Let $G$ be a finite abelian group acting faithfully and linearly on a smooth hypersurface of degree $d$ in the complex projective space $\mathbb{P}^{N-1}$. Suppose $G\subset PGL(N, \mathbb{C})$ can be lifted to a subgroup of $GL(N,\mathbb{C})$. Suppose moreover that there exists an element $g$ in $G$ such that $G/\langle g\rangle$ has order coprime to $d-1$. Then all possible $G$ are determined (Theorem 4.3). As an application, we derive (Theorem 4.8) all possible orders of linear automorphisms of smooth hypersurfaces for any given $(d,N)$. In particular, …

Dynamical degrees of automorphisms on abelian varieties

For any given Salem number, we construct an automorphism on a simple abelian variety whose first dynamical degree is the square of the Salem number. Our construction works for both simple abelian varieties with totally indefinite quaternion multiplication and for simple abelian varieties of the second kind. We then give a complete classification of the dynamical degree sequences for abelian varieties of dimension at most four and obtain an ergodic result for sequences of pullbacks of forms.

Equivalence of Group Actions on Riemann Surfaces

We produce for each natural number $n \geq 3$ two 1--parameter families of Riemann surfaces admitting automorphism groups with two cyclic subgroups $H_{1}$ and $H_{2}$ of orden $2^{n}$, that are conjugate in the group of orientation--preserving homeomorphism of the corresponding Riemann surfaces, but not conjugate in the group of conformal automorphisms. This property implies that the subvariety $\mathcal{M}_{g}(H_{1})$ of the moduli space $\mathcal{M}_{g}$ consisting of the points representing the Riemann surfaces of genus $g$ admitting a group of automorphisms topologically conjugate to $H_{…

Non-Abelian Class Field Theory for Riemann Surfaces

In this paper, using what we call a micro reciprocity law, we complete Weil's program for non-abelian class field theory of Riemann surfaces.

Automorphisms of hyperkähler manifolds in the view of topological entropy

First we show that any group of automorphisms of null-entropy of a projective hyperkähler manifold $M$ is almost abelian of rank at most $ρ(M) - 2$. We then characterize automorphisms of a K3 surface with null-entropy and those with positive entropy in algebro-geometric terms. We also give an example of a group of automorphisms which is not almost abelian in each dimension.

An elementary abelian $p$-cover of the Hermitian curve with many automorphisms

The full automorphism group of a certain elementary abelian $p$-cover of the Hermitian curve in characteristic $p>0$ is determined. It is remarkable that the order of Sylow $p$-groups of the automorphism group is close to Nakajima's bound in terms of the $p$-rank. Weierstrass points, Galois points, Frobenius nonclassicality, and arc property are also investigated.

Abelian automorphism groups of threefolds of general type

This thesis is devoted to the study of abelian automorphism groups of surfaces and $3$-folds of general type over complex number field $\Bbb C$. We obtain a linear bound in $K^3$ for abelian automorphism groups of $3$-folds of general type whose canonical divisor $K$ is numerically effective, and we improve on Xiao's results on abelian automorphism groups of minimal smooth projective surfaces of general type. More precisely, the main results in this thesis are the following. {\bf Theorem 3.0.} Let $X$ be a smooth 3-fold of general type over the complex number field, $K$ the canonical divisor o…

A bound for the number of automorphisms of an arithmetic Riemann surface

We show that for every g > 1 there is a compact arithmetic Riemann surface of genus g with at least 4(g-1) automorphisms, and that this lower bound is attained by infinitely many genera, the smallest being 24.

Linear bound for abelian automorphisms of varieties of general type

We prove: Let $G$ be a finite abelian group acting faithfully on a complex smooth project variety $X$ of general type with numerically effective canonical divisor, of dimension $n$. Then $$|G| \le C(n)K_X^n ,$$ where $C(n)$ depends only on $n$.

Automorphism groups of pseudoreal Riemann surfaces

A smooth complex projective curve is called pseudoreal if it is isomorphic to its conjugate but is not definable over the reals. Such curves, together with real Riemann surfaces, form the real locus of the moduli space $\mathcal M_g$. This paper deals with the classification of pseudoreal curves according to the structure of their automorphism group. We follow two different approaches existing in the literature: one coming from number theory, dealing more generally with fields of moduli of projective curves, and the other from complex geometry, through the theory of NEC groups. Using the first…

Salem Numbers and Abelian Surface Automorphisms

We classify two-dimensional complex tori admitting automorphisms with positive entropy in terms of the entropies they exhibit. For each possible positive value of entropy, we describe the set of two-dimensional complex tori admitting automorphisms with that entropy.

拟阿贝尔群作为黎曼曲面的自同构群

Quasi-abelian group as automorphism group of Riemann surfaces

本文研究拟阿贝尔群 $QA_n$ 在闭黎曼曲面、伪实黎曼曲面和闭克莱因曲面上的共形/反共形作用，解决了其最小亏格问题，并分别讨论了群含反共形元素与仅含共形元素两种情形下的拓扑刚性作用问题。

双曲格的虚拟阿贝尔对称群

On hyperbolic lattices whose reduced automorphism group is virtually abelian

本文分类了对称群为虚拟阿贝尔群的整数格，进而完成虚拟阿贝尔自同构群 K3 曲面的分类，并在附录给出正交群弱逼近算法以判定不定格等距性。

Automorphism Groups of Cyclic p-gonal Pseudo-real Riemann Surfaces

In this article we prove that the full automorphism group of a cyclic $p$-gonal pseudo-real Riemann surface of genus $g$ is either a semidirect product $C_{n}\ltimes C_{p}$ or a cyclic group, where $p$ is a prime $>2$ and $g>(p-1)^{2}$. We obtain necessary and sufficient conditions for the existence of a cyclic $p$-gonal pseudo-real Riemann surface with full\ automorphism group isomorphic to a given finite group. Finally we describe some families of cyclic $p$-gonal pseudo-real Riemann surfaces where the order of the full automorphism group is maximal and show that such families determine some…

Reduced norms and the Riemann-Roch theorem for Abelian varieties

We explain how the Riemann-Roch theorem for divisors on an abelian variety $A$ is related to the reduced norms of the Wedderburn components of $\operatorname{End}^0(A)$ the $\mathbb{Q}$-endomorphism algebra of $A$. We then describe consequences and examples.

Deformation of Curves with Automorphisms and representations on Riemann-Roch spaces

We study the deformation theory of nonsigular projective curves defined over algebraic closed fields of positive characteristic. We show that under some assumptions the local deformation problem for automorphisms of powerseries can be reduced to a deformation problem for matrix representations. We study both equicharacteristic and mixed deformations in the case of two dimensional representations.

Tropical Theta Functions and Riemann-Roch Inequality for Tropical Abelian Surfaces

We show that the space of theta functions on tropical tori is identified with a convex polyhedron. We also show a Riemann-Roch inequality for tropical abelian surfaces by calculating the self-intersection numbers of divisors.

Riemann-Roch, Stability and New Non-Abelian Zeta Functions for Number Fields

In this paper, we introduce a geometrically stylized arithmetic cohomology for number fields. Based on such a cohomology, we define and study new yet genuine non-abelian zeta functions for number fields, using an intersection stability.

A classification of the automorphism groups of polarized abelian threefolds over finite fields

We give a classification of maximal elements of the set of finite groups that can be realized as the automorphism groups of polarized abelian threefolds over finite fields.

阿贝尔簇的自同构族

Families of automorphisms of abelian varieties

本文研究复单位圆盘参数化的极化阿贝尔簇族上自同构动力学的代数性质，证明无分圆因子时映射可正则化当且仅当族不退化，并证明平移族总可正则化，同时刻画了轨道闭包。

On the Group of Automorphisms of Cyclic Covers of the Riemann Sphere

This note uses some recent calculations of Conder and Bujalance (on classifying finite index group extensions of Fuchsian groups with abelian quotient and torsion free kernel) in order to determine the full automorphism groups of some cylic coverings of the line (including curves of Fermat and Lefschetz type). The answer is complete for cyclic covers that branch over three points.

Linearisation of finite abelian subgroups of the Cremona group of the plane

This article gives the proof of results announced in [J. Blanc, Finite Abelian subgroups of the Cremona group of the plane, C.R. Acad. Sci. Paris, Sér. I 344 (2007), 21-26.] and some description of automorphisms of rational surfaces. Given a finite Abelian subgroup of the Cremona group of the plane, we provide a way to decide whether it is birationally conjugate to a group of automorphisms of a minimal surface. In particular, we prove that a finite cyclic group of birational transformations of the plane is linearisable if and only if none of its non-trivial elements fix a curve of positive gen…

Action of Correspondences on Filtrations on Cohomology and 0-cycles of Abelian Varieties

We prove that, given a symmetrically distinguished correspondence of a suitable complex abelian variety (which include any abelian variety of dimension atmost 5, powers of complex elliptic curves, etc.) which vanishes as a morphism on a certain quotient of its middle singular cohomology, then it vanishes as a morphism on the deepest part of a particular filtration on the Chow group of 0-cycles of the abelian variety. As a consequence, we prove that given an automorphism of such an abelian variety, which acts as the identity on a certain quotient of its middle singular cohomology, then it acts …

具有 Weierstrass 点传递作用的无穷多 Riemann 曲面

Infinitely many Riemann surfaces with transitive action on Weierstrass points

本文证明了存在无穷多个两两不同构的非超椭圆 Riemann 曲面，其自同构群在 Weierstrass 点上传递作用，并在假设自同构群为单群时找出了所有满足该条件的紧 Riemann 曲面。

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties. In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group $G$ there exist infinitely many components $M$ of the moduli of varieties with ample canonical class such that the generic automorphism group $G_M$ is equal to $G$. In order to construct the examples, we use abelian covers, i.e. Galois cover whose Galois group is …