Non-Abelian Class Field Theory for Riemann Surfaces

Abelian pole systems and Riemann-Schottky type systems

In this survey of works on a characterization of Jacobians and Prym varieties among indecomposable principally polarized abelian varieties via the soliton theory we focus on a certain circle of ideas and methods which show that the characterization of Jacobians as ppav whose Kummer variety admits a trisecant line and the Pryms as ppav whose Kummer variety admits a pair of symmetric quadrisecants can be seen as an abelian version of pole systems arising in the theory of elliptic solutions to the basic soliton hierarchies. We present also recent results in this direction on the characterization …

非阿贝尔霍奇理论的渐近几何与黎曼–希尔伯特对应，秩三 $\widetilde{E}_6$ 情形

Asymptotic geometry of non-abelian Hodge theory and Riemann--Hilbert correspondence, rank three $\widetilde{E}_6$ case

本文在三孔球面上证明秩三的几何 P=W 猜想，通过渐近阿贝尔化与等变方法给出相关映射的 WKB 展开，并分析其 Stokes 现象，从而刻画特征簇在无穷远处的拓扑。

Riemann Hypothesis for Non-Abelian Zeta Functions of Curves over Finite Fields

In this paper, we develop some basic techniques towards the Riemann hypothesis for higher rank non-abelian zeta functions of an integral regular projective curve of genus $g$ over a finite field $\mathbb F_q$. As an application of the Riemann hypothesis for these genuine zeta functions, we obtain some explicit bounds on the fundamental non-abelian $α$- and $β$-invariants of $X/\mathbb F_q$ in terms of $X$ and $n,\, q$ and $g$: $$α_{X,\mathbb F_q;n}(mn) = \sum_{V}\frac{q^{h^0(X,V)}-1}{\#\mathrm{Aut}(V)} \qquad{\rm and}\qquad β_{X,\mathbb F_q;n}(mn ):= \sum_{V}\frac{1}{\#{\mathrm Aut}(V)}\qquad(…

Moments on Riemann surfaces and hyperelliptic Abelian integrals

In the present paper we solve the following different but interrelated problems: (a) the moment problem on Riemann surfaces, (b) the vanishing problem of polynomial Abelian integrals of dimension zero on the projective plane, (c) the vanishing problem of polynomial hyperelliptic Abelian integrals.

亏格2曲线的非阿贝尔zeta函数的黎曼猜想

Riemann Hypothesis for Non-Abelian Zeta Functions of Genus 2 Curves

本文针对有限域上亏格2曲线的Weng非阿贝尔zeta函数，利用半稳定丛模空间几何性质、高秩结果与渐近分析，证明了当秩n趋于无穷时黎曼猜想成立。

黎曼曲面的非阿贝尔方法第一部分：朗斯基几何

A Non-Abelian Approach to Riemann Surfaces Part I: Wronskian Geometry

针对黎曼曲面上的射影平坦全纯向量丛开展研究，通过为其分配Wronskian线丛、定义亚纯截面除法，将Abel恒等式解释为对应线丛的第一陈类。

非阿贝尔方法在黎曼面中的应用

A Non-Abelian Approach to Riemann Surfaces

本文提出非阿贝尔的规范框架，用于黎曼面上的斯瓦茨导数和二阶微分方程，扩展了狄德金方法至一般一参数曲线族，并讨论了三维三次三重面的周期和机械中的质量-弹簧系统。

具有阿贝尔对称性的黎曼曲面的等对称轨迹的边界

Boundary of equisymmetric loci of Riemann surfaces with abelian symmetry

本文研究了亏格$g\geq 2$黎曼曲面模空间中具有阿贝尔自同构群的等对称轨迹，确定了其边界最大维数层的拓扑类型，并用具有固定边数的树描述了超椭圆和循环$p$-边作用边界处的所有拓扑层。

Non-abelian reciprocity laws on a Riemann surface

On a Riemann surface there are relations among the periods of holomorphic differential forms, called Riemann's relations. If one looks carefully in Riemann's proof, one notices that he uses iterated integrals. What I have done in this paper is to generalize these relations to relations among generating series of iterated integrals. Since the main result is formulated in terms of generating series, it gives infinitely many relations - one for each coefficient of the generating series. The lower order terms give the well known classical relations. The new result is reciprocity for the higher deg…

Riemann reciprocity in higher dimensions

The reciprocity law for abelian differentials of first and second kind is generalized to higher-dimensional varieties. It is shown that $H^1(V)$ of a polarized variety $V$ is encoded in the Laurent data along a curve germ in $V$, with the polarization form on $H^1(V)$ corresponding to the {\em one-dimensional} residue pairing. This associates an {\em extended abelian variety} to $V$; if $V$ is an abelian variety itself, our construction ``extends" it, even when $V$ is not a Jacobian.

Higher adeles and non-abelian Riemann-Roch

We show a Riemann-Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson-Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The theorem is obtained by combining a group ring coefficient version of the local Riemann-Roch formula as in Kapranov-Vasserot with results on K-groups of group rings and an explicit description of group ring bundles over P^1. Our set-up provides an extension of several aspects of the classical Fr"ohlich theory of the Galois module structure of rings of integers of …

New Non-Abelian Zeta Functions for Curves over Finite Fields

In this paper, we introduce and study two new types of non-abelian zeta functions for curves over finite fields, which are defined by using (moduli spaces of) semi-stable vector bundles and non-stable bundles. A Riemann-Weil type hypothesis is formulated for zeta functions associated to semi-stable bundles, which we think is more canonical than the other one. All this is motivated by (and hence explains in a certain sense) our work on non-abelian zeta functions for number fields.

On the Geometry of Flat Pseudo-Riemannian Homogeneous Spaces

Let $M$ be complete flat pseudo-Riemannian homogeneous manifold and $Γ\subset\Iso(\RR^n_s)$ its fundamental group. We show that $M$ is a trivial fiber bundle $G/Γ\to M\to\RR^{n-k}$, where $G$ is the Zariski closure of $Γ$ in $\Iso(\RR^n_s)$. Moreover, we show that the $G$-orbits in $\RR^n_s$ are affinely diffeomorphic to $G$ endowed with the (0)-connection. If the induced metric on the $G$-orbits is non-degenerate, then $G$ (and hence $Γ$) has linear abelian holonomy. If additionally $G$ is not abelian, then $G$ contains a certain subgroup of dimension 6. In particular, for non-abelian $G$ orb…

What is the Jacobian of a Riemann surface with boundary?

We define the Jacobian of a Riemann surface with analytically parametrized boundary components. These Jacobians belong to a moduli space of ``open abelian varieties'' which satisfies gluing axioms similar to those of Riemann surfaces, and therefore allows a notion of ``conformal field theory'' to be defined on this space. We further prove that chiral conformal field theories corresponding to even lattices factor through this moduli space of open abelian varieties.

Degenerate abelian function fields

Originally, an abelian function field is the field of meromorphic functions on the Jacobi variety J(X) of a compact Riemann surface X. It is generated by the fundamental abelian functions belonging to the meromorphic function field on X. We study this relation for singular curves.

Perverse sheaves on Riemann surfaces as Milnor sheaves

Constructible sheaves of abelian groups on a stratified space can be equivalently described in terms of representations of the exit-path category. In this work, we provide a similar presentation of the abelian category of perverse sheaves on a stratified surface in terms of representations of the so-called paracyclic category of the surface. The category models a hybrid exit-entrance behaviour with respect to chosen sectors of direction, placing it "in between" exit and entrance path categories. In particular, this perspective yields an intrinsic definition of perverse sheaves as an abelian ca…

General Uniformity of Zeta Functions

Using analytic torsion associated to stable bundles, we introduce zeta functions for compact Riemann surfaces. To justify the well-definedness, we analyze the degenerations of analytic torsions at the boundaries of the moduli spaces, the singularities of analytic torsions at Brill-Noether loci, and the asymptotic behaviors of analytic torsions with respect to the degree. These new yet intrinsic zetas, both abelian and non-abelian, are expected to play key roles to understand global analysis and geometry of Riemann surfaces, such as the Tamagawa number conjecture for Riemann surfaces, searched …

Universal moduli spaces of Riemann surfaces

We construct a moduli space for Riemann surfaces that is universal in the sense that it represents compact Riemann surfaces of any finite genus. This moduli space is stratifed according to genus, and it carries a metric and a measure that induce a Riemannian metric and a finite volume measure on each stratum. Applications to the Plateau-Douglas problem for minimal surfaces of varying genus and to the partition function of Bosonic string theory are outlined. The construction starts with a universal moduli space of Abelian varieties. This space carries a structure of an infinite dimensional loca…

On Spaces Associated with Invariant Divisors on Galois Covers of Riemann Surfaces and Their Applications

Let $f:X \to S$ be a Galois cover of Riemann surfaces, with Galois group $G$. In this paper we analyze the $G$-invariant divisors on $X$, and their associated spaces of meromorphic functions, differentials, and $q$-differentials. We generalize the trace formula for non-trivial elements of $G$ on $q$-differentials, as well as the Chevalley--Weil Formula. When $G$ is Abelian or when the genus of $S$ is 0 we prove additional results, and we also determine the non-special $G$-invariant divisors when both conditions are satisfied.

Non-Archimedean and tropical theta functions

We define a tropicalization procedure for theta functions on abelian varieties over a non-Archimedean field. We show that the tropicalization of a non-Archimedean theta function is a tropical theta function, and that the tropicalization of a non-Archimedean Riemann theta function is a tropical Riemann theta function, up to scaling and an additive constant. We apply these results to the construction of rational functions with prescribed behavior on the skeleton of a principally polarized abelian variety. We work with the Raynaud--Bosch--Lütkebohmert theory of non-Archimedean theta functions for…

Finiteness criteria and uniformity of integral sections in some families of abelian varieties

Let $A$ be abelian variety over the function field $K$ of a compact Riemann surface $B$. Fix a model $f \colon \mathcal{A} \to B$ of $A/K$ and a certain effective horizontal divisor $\DD \subset \mathcal{A}$. We give a sufficient condition on the divisor $\DD$ for the finiteness of the set of $(S, \DD)$-integral sections for every finite subset $S \subset B$. These integral sections $σ$ correspond to rational points in $A(K)$ which satisfy the geometric condition $f ( σ(B) \cap \DD)\subset S$. This notion is the geometric variant of integral solutions of a system of \emph{Diophantine equations…

Topologically unique maximal elementary Abelian group actions on compact oriented surfaces

We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus $σ\geq 2$ which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.

Universal Hodge Bundle and Mumford Isomorphisms on the Abelian Inductive Limit of Teichmüller Spaces

Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed (genus-independent) tensor powers of the Hodge line bundle on $\Mg$. Here we obtain a coherent, genus-independent description of the Mumford isomorphisms over certain infinite-dimensional ``universal'' parameter spaces of compact Riemann surfaces (with or without marked points). We work with an inductive limit of Teichmüller spaces comprising complex structures on a certain …

The arithmetic volume of the moduli space of abelian surfaces

Let $\mathcal{A}_g$ denote the moduli stack of principally polarized abelian varieties of dimension $g$. The arithmetic height, or arithmetic volume, of $\overline{\mathcal{A}}_g$, is defined to be the arithmetic degree of the metrized Hodge bundle $\overlineω_g$ on $\overline{\mathcal{A}}_g$. In 1999, Kühn proved a formula for the arithmetic volume of $\overline{\mathcal{A}}_1$ in terms of special values of the Riemann zeta function. In this article, we generalize his result to the case $g=2$.

A dynamical approach to generalized Weil's Riemann hypothesis and semisimplicity

Let $X$ be a smooth projective variety over an algebraically closed field of arbitrary characteristic, and $f$ a dynamical correspondence of $X$. In 2016, the second author conjectured that the dynamical degrees of $f$ defined by the pullback actions on étale cohomology groups and on numerical cycle class groups are equivalent, which we call the dynamical degree comparison (DDC) conjecture. It contains the generalized Weil's Riemann hypothesis (for polarized endomorphisms) as a special case. To proceed, we introduce the so-called Conjecture $G_r$, which is a quantitative strengthening of the s…

Supersymmetric partition functions on Riemann surfaces

We present a compact formula for the supersymmetric partition function of 2d N=(2,2), 3d N=2 and 4d N=1 gauge theories on $Σ_g \times T^n$ with partial topological twist on $Σ_g$, where $Σ_g$ is a Riemann surface of arbitrary genus and $T^n$ is a torus with n=0,1,2, respectively. In 2d we also include certain local operator insertions, and in 3d we include Wilson line operator insertions along $S^1$. For genus g=1, the formula computes the Witten index. We present a few simple Abelian and non-Abelian examples, including new tests of non-perturbative dualities. We also show that the large N par…

Non-Abelian Simple Groups Act with Almost All Signatures

The topological data of a group action on a compact Riemann surface is often encoded using a tuple $(h;m_1,\dots ,m_s)$ called its signature. There are two easily verifiable arithmetic conditions on a tuple necessary for it to be a signature of some group action. In the following, we derive necessary and sufficient conditions on a group $G$ for when these arithmetic conditions are in fact sufficient to be a signature for all but finitely many tuples that satisfy them. As a consequence, we show that all non-Abelian finite simple groups exhibit this property.

Integrable linear equations and the Riemann-Schottky problem

We prove that an indecomposable principally polarized abelian variety $X$ is the Jacobain of a curve if and only if there exist vectors $U\neq 0,V$ such that the roots $x_i(y)$ of the theta-functional equation $θ(Ux+Vy+Z)=0$ satisfy the equations of motion of the {\it formal infinite-dimensional Calogero-Moser system}

Unipotent Schottky bundles on Riemann surfaces and complex tori

We study a natural map from representations of a free (resp. free abelian) group of rank g in GL_r(C), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (resp. complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor induces an equivalence between the category of unipotent representations of Schottky groups and the category of unipotent vector bundles on X. We also show that, over a complex torus, any vector or principal bundle with a flat holomorphic connection is Schottky.

The sigma function for Weierstrass semigroups <3,7,8> and <6,13,14,15,16>

Compact Riemann surfaces and their abelian functions are instrumental to solve integrable equations; more recently the representation theory of the Monster and related modular form have pointed to the relevance of $τ$-functions, which are in turn connected with a specific type of abelian function, the (Kleinian) $σ$-function. This paper proposes a construction of $σ$-functions based on the nature of the Weierstrass semigroup at one point of the Riemann surface as a generalization of the construction of plane affine models of the Riemann surface. Because our definition is algebraic, we are able…

非阿贝尔霍奇理论与Higgs丛的模空间

Non-Abelian Hodge Theory and Moduli Spaces of Higgs Bundles

本文研究非阿贝尔霍奇理论与Higgs丛模空间，探讨了Hitchin纤维的几何性质及拓扑镜像对称性，并通过计算庞加莱多项式验证了相关对称性。

非阿贝尔 Hodge 对应的温和导引

A Gentle Introduction to the Non-Abelian Hodge Correspondence

本文以教学方式介绍非阿贝尔 Hodge 对应，阐明基本群表示与全纯丛之间的联系，并梳理其核心定理、几何结构与相关应用。

Principal Schottky Bundles over Riemann surfaces

We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky $G$-bundles have trivial topological type. Generalizing the Schottky moduli map introduced in Florentino to the setting of principal bundles, we prove its local surjectivity at the good and unitary locus. Finally, we prove that the Schottky map is surjective onto the space of flat bundles for two special classes: when G…

椭圆曲线上第二类阿贝尔微分的等周期形变与Boussinesq方程

Isoperiodic deformations of Abelian differentials of the second kind over elliptic curves and the Boussinesq equation

研究保持周期不变的黎曼曲面和第二类阿贝尔微分的形变，导出控制极点位置的有理系数二阶常微分方程，并应用于Boussinesq方程的亏格一解。