Modular forms of degree two and curves of genus two in characteristic two

Hilbert模形式中零 genus 曲线的部分Hassé不变量

Partial Hasse invariants for genus zero curves in Hilbert modular varieties

本文研究了Hilbert模形式中零 genus 曲线的部分Hassé不变量的特征零提升，并通过计算Teichmüller曲线得到类似Deuring的公式。

关于亏格3或4的Siegel模簇的超奇异轨迹

On the supersingular locus of the Siegel modular variety of genus 3 or 4

本文研究亏格3或4的Siegel模簇的超奇异轨迹，在perfection后将其分解为Coxeter型Deligne-Lusztig簇与有限维仿射空间乘积的不交并。

Covariants of binary sextics and vector-valued Siegel modular forms of genus two

We extend Igusa's description of the relation between invariants of binary sextics and Siegel modular forms of degree two to a relation between covariants and vector-valued Siegel modular forms of degree two. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree two.

Finiteness results for modular curves of genus at least 2

A curve X over the field Q of rational numbers is modular if it is dominated by X_1(N) for some N; if in addition the image of its jacobian in J_1(N) is contained in the new subvariety of J_1(N), then X is called a new modular curve. We prove that for each integer g at least 2, the set of new modular curves over Q of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de Franchis-Severi Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of the new part …

从亏格2的模方程计算同源

Computing isogenies from modular equations in genus two

提出一种算法，利用模方程和显式Kodaira-Spencer同构，在特征足够大且曲线足够一般的条件下，显式计算亏格2曲线上Jacobian的l-同源或循环核同源。

Algebraic cycles on genus two modular fourfolds

We study universal families of stable genus two curves with level structure. Among other things, it is shown that the (1,1) part is spanned by divisor classes, and that there are no cycles of type (2,2) in the third cohomology of the first direct image. Using this, we deduce the Hodge and Tate conjectures hold for these varieties.

Modular Curves Of Genus 2

We prove that there is only a finite number of genus 2 curves C defined over Q such that there exists a nonconstant morphism pi:X_1(N) --->C defined over Q and the jacobian of C, J(C), is a Q-factor of the new part of the jacobian of X_1(N), J_1(N)^{new}. Moreover, we prove that there are only 149 genus two curves of this kind with the additional requeriment that their jacobians are Q-simple. We determine the corresponding newforms and present equations for all these curves.

Genus two curves with quaternionic multiplication and modular jacobian

We describe a method to determine all the isomorphism classes of principal polarizations of the modular abelian surfaces $A_f$ with quaternionic multiplication attached to a normalized newform $f$ without complex multiplication. We include an example of $A_f$ with quaternionic multiplication for which we find numerically a curve $C$ whose Jacobian is $A_f$ up to numerical approximation, and we prove that it has quaternionic multiplication and is isogenous to $A_f$.

Computing genus 2 Hilbert-Siegel modular forms over $\Q(\sqrt{5})$ via the Jacquet-Langlands correspondence

In this paper we present an algorithm for computing Hecke eigensystems of Hilbert-Siegel cusp forms over real quadratic fields of narrow class number one. We give some illustrative examples using the quadratic field $\Q(\sqrt{5})$. In those examples, we identify Hilbert-Siegel eigenforms that are possible lifts from Hilbert eigenforms.

Invariants of models of genus one curves and modular forms

An invariant of a model of genus one curve is a polynomial in the coefficients of the model that is stable under certain linear transformations. The classical example of an invariant is the discriminant, which characterizes the singularity of models. The ring of invariants of genus one models over a field is generated by two elements. Fisher normalized these invariants for models of degree n=2,3,4 in such a way that these invariants are moreover defined over the integers. We provide an alternative way to express these normalized invariants using modular forms. This method relies on a direct co…

dormant模曲线的亏格公式及其函数域的渐进行为

Genus formulas for dormant modular curves and asymptotic behavior of their function fields

本文研究了基于高阶水平dormant PGL_2-opers的模曲线的亏格公式，并分析了其函数域的渐进行为，与经典模曲线和Drinfeld模曲线进行比较。

Concomitants of Ternary Quartics and Vector-valued Siegel and Teichmüller Modular Forms of Genus Three

We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichmüller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichmüller cusp forms on \overline{M}_g and the middle cohomology of symplectic local systems on M_g. In genus 3, we make this explicit in a large number of cases.

非超椭圆模曲线的亏格3

Non-hyperelliptic modular curves of genus 3

本文研究了亏格为3的非超椭圆模曲线的模性，提出了一种判定其模性的条件和计算其方程的算法，并探讨了新的模曲线的定义。

Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields

We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize decomposable principally polarized abelian surfaces, they are also moduli spaces for genus-2 curves covering elliptic curves via a map of fixed degree. We thereby extend classical work of Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska, Völklein, Magaard and others, producing explicit families of reducible Jacobians. In particular, we produce a birational model for the modul…

Computing Genus 1 Jacobi Forms

We develop an algorithm to compute Fourier expansions of vector valued modular for Weil representations. As an application, we compute explicit linear equivalences of special divisors on modular varieties of orthogonal type. We define three families of Hecke operators for Jacobi forms, and analyze the induced action on vector valued modular forms. The newspaces attached to one of these families are used to give a more memory efficient version of our algorithm.

Bielliptic modular curves $X_0^*(N)$ with square-free levels

Let $N\geq 1$ be a square-free integer such that the modular curve $X_0^*(N)$ has genus $\geq 2$. We prove that $X_0^*(N)$ is bielliptic exactly for $19$ values of $N$, and we determine the automorphism group of these bielliptic curves. In particular, we obtain the first examples of nontrivial $Aut(X_0^*(N))$ when the genus of $X_0^*(N)$ is $\geq 3$. Moreover, we prove that the set of all quadratic points over $\mathbb{Q}$ for the modular curve $X_0^*(N)$ with genus $\geq 2$ and $N$ square-free is not finite exactly for $51$ values of $N$.

Elliptic genus and modular differential equations

We study modular differential equations for the basic weak Jacobi forms in one abelian variable with applications to the elliptic genus of Calabi--Yau varieties. We show that the elliptic genus of any $CY_3$ satisfies a differential equation of degree one with respect to the heat operator. For a $K3$ surface or any $CY_5$ the degree of the differential equation is $3$. We prove that for a general $CY_4$ its elliptic genus satisfies a modular differential equation of degree $5$. We give examples of differential equations of degree two with respect to the heat operator similar to the Kaneko--Zag…

亏格二拟Siegel模形式与环面Calabi-Yau三流形的Gromov-Witten理论

Genus Two Quasi-Siegel Modular Forms and Gromov-Witten Theory of Toric Calabi-Yau Threefolds

本文发展了亏格二拟Siegel模形式与拟Siegel Jacobi形式的微分环理论，并将其应用于带膜的局部Calabi-Yau三流形的拓扑递归，证明了开闭Gromov-Witten势本质上是亏格二拟Siegel模形式。

Basic vector valued Siegel modular forms of genus two

We study over rings of scalar valued Siegel modular forms. modules of vector valued modular forms of degree two. For the two simplest representations, standard and Sym^2, appears rather natural consider the cases of the group $Γ[4,8] $ and $Γ[2,4].$ In these case we give the complete structure of the modules . The main tools are Theta functions and their derivatives evaluated at $z=0$

The Kodaira dimension of Siegel modular varieties of genus 3 or higher

We consider the moduli space $A_{pol}(n)$ of (non-principally) polarised abelian varieties of genus $g\geq3$ with coprime polarisation and full level-$n$ structure. Based upon the analysis of the Tits building in math/0405321, we give an explicit lower bound on $n$ that is sufficient for the compactified moduli space to be of general type if one further explicit condition is satisfied.

Siegel modular forms of genus 2 attached to elliptic curves

The object of this article is to construct certain classes of arithmetically significant, holomorphic Siegel cusp forms F of genus 2, which are neither of Saito-Kurokawa type, in which case the degree 4 spinor L-function L(s, F) is divisible by an abelian L-function, nor of Yoshida type, in which case L(s,F) is a product of L-series of a pair of elliptic cusp forms. Two key examples are the forms F defined by the symmetric cube of a non-CM elliptic curve E over the rationals, and those defined by anticyclotomic twists (of even non-zero weight) of the base change of such an E to an imaginary qu…

关于亏格1曲线的亏格3覆盖的Prym簇

On the Prym variety of genus 3 covers of genus 1 curves

本文针对亏格1曲线的非超椭圆亏格3覆盖，构造显式亏格2曲线，使雅可比簇同源于原曲线与覆盖曲线的雅可比簇乘积，推广了Nils Bruin的结果。

Symmetries of genus zero modular operad

In this article combining survey and certain research results, we introduce a categorical framework for description of symmetries of genus zero modular operad. This description merges the techniques of recent "persistence homology" studies and the classical formalism of groupoids. We show that the contravariant "poset in groupoids" embodying these symmetries, provides a new avatar of profinite Grothendieck-Teichmüller group acting upon this operad but seemingly not related with representations of the Galois group of all algebraic numbers.

Quadratic Chabauty for modular curves: Algorithms and examples

We describe how the quadratic Chabauty method may be applied to explicitly determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell--Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or nontrivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin--Lehner quotients $X_0^+(N…

Two-coverings of Jacobians of curves of genus two

We find explicit equations for two-coverings of Jacobians of genus two curves over an arbitrary ground field of characteristic different from two.

高亏格 Szegö 核的循环乘积、模张量与多对数

Cyclic products of higher-genus Szegö kernels, modular tensors and polylogarithms

本文研究任意亏格 Riemann 曲面上多个 Szegö 核的循环乘积，将其分解为承载自旋结构依赖的模张量与由高亏格多对数积分核构成的系数，并求出相应反全纯模导数。

Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms

In the paper we study two types of relations: a one is between the elliptic genus of Calabi-Yau manifolds and Jacobi modular forms, another one is between the second quantized elliptic genus, Siegel modular forms and Lorentzian Kac-Moody Lie algebras. We also determine the structure of the graded ring of the weak Jacobi forms with integral Fourier coefficients. It gives us a number of applications to the theory of elliptic genus and of the second quantized elliptic genus.

Optimal curves of genus 1,2 and 3

In this survey, we discuss the problem of the maximum number of points of curves of genus 1,2 and 3 over finite fields

Genus one half stacky curves violating the local-global principle

For any number field, we prove that there exists a stacky curve of genus one half defined over the ring of its integers violating the local-global principle for integral points.

A database of nonhyperelliptic genus 3 curves over Q

We report on the construction of a database of nonhyperelliptic genus 3 curves over Q of small discriminant.

A census of all genus 4 curves over the field with 2 elements

We explain how we computed equations for all genus 4 curves defined of the field with 2 elements, up-to-isomorphism, and some of the data we obtained. We give descriptions also of nice models for genus 4 curves over characteristic 2 fields, in both the hyperelliptic case and the trigonal case.

Curves of genus 3 over small finite fields

We present a table containing the maximal number of rational points on a genus 3 curve over a field of cardinality q, for all q<100. Also, some remarks on Frobenius non-classical quartics over finite fields are given.

The Number of Rational Points On Genus 4 Hyperelliptic Supersingular Curves in Characteristic 2

One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: Which numbers occur as the number of points on a curve of genus $g$? The same question can be asked of various subclasses of curves. In this article we classify the possibilities for the number of points on genus 4 hyperelliptic supersingular curves over finite fields of order $2^n$, $n$ odd.

Genus-2 curves and Jacobians with a given number of points

We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of F-rational points. In the case of the Jacobian, we show that any `CM-construction' to produce the required genus-2 curves necessarily takes time exponential in the size of its input. On the other hand, we provide an algorithm for producing a genus-2 curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing…

A Universal Genus-Two Curve from Siegel Modular Forms

Let $\mathfrak p$ be any point in the moduli space of genus-two curves $\mathcal M_2$ and $K$ its field of moduli. We provide a universal equation of a genus-two curve $\mathcal C_{α, β}$ defined over $K(α, β)$, corresponding to $\mathfrak p$, where $α$ and $β$ satisfy a quadratic $α^2+ b β^2= c$ such that $b$ and $c$ are given in terms of ratios of Siegel modular forms. The curve $\mathcal C_{α, β}$ is defined over the field of moduli $K$ if and only if the quadratic has a $K$-rational point $(α, β)$. We discover some interesting symmetries of the Weierstrass equation of $\mathcal C_{α, β}$. …

Calabi-Yau簇椭圆亏格的极小阶模微分方程

Modular differential equations of minimal orders of the elliptic genus of Calabi--Yau varieties

本文研究Calabi-Yau簇椭圆亏格的模微分方程，证明严格六维情形最小阶为四，并为六维超Kähler簇构造7阶方程，刻画非一般解的结构。

亏格二稳定映射、局部方程与模消解

Genus Two Stable Maps, Local Equations and Modular Resolutions

本研究通过对参数化亏格二预稳定对的阿廷叠进行模胀开构造，局部对角化相关稳定映射模空间的导出对象，消解主分支奇点并使全空间仅含正规交叉奇点，方法有望推广至高亏格。

Modularity and uniformization of a higher genus algebraic space curve, its distinct arithmetical realizations by cohomology groups and $E_6$, $E_7$, $E_8$-singularities

We prove the modularity for an algebraic space curve $Y$ of genus $50$ in $\mathbb{P}^5$, which consists of $21$ quartic polynomials in six variables, by means of an explicit modular parametrization by theta constants of order $13$. This provides an example of modularity, explicit uniformization and hyperbolic uniformization of arithmetic type for a higher genus algebraic space curve. In particular, it gives a new example for Hilbert's 22nd problem. This gives $21$ modular equations of order $13$, which greatly improve the result of Ramanujan and Evans on the construction of modular equations …

On a ring of modular forms related to the Theta gradients map in genus 2

The level moduli space $A_g^{4,8}$ is mapped to the projective space by means of gradients of odd Theta functions, such a map turning out no to be injective in the genus 2 case. In this work a congruence subgroup $Γ$ is located between $Γ_2(4,8)$ and $Γ_2(2,4)$ in such a way the map factors on the related level moduli space $A_Γ$, the new map being injective on $A_Γ$. Satake's compactification $\text{Proj}A(Γ)$ and the desingularization $\text{Proj}S(Γ)$ are also due to be investigated, since the map does not extend to the boundary of the compactification; to aim at this, an algebraic descript…

Genus zero modular operad and absolute Galois group

In this article, we develop the geometry of canonical stratifications of the spaces $\mathcal{M}_{0,n}$ and prepare ground for studying the action of the Galois group $Gal (\overline{\mathbb{Q}} /\mathbb{Q})$ upon strata. We define and introduce a version of a gravity operad constructed for a class of moduli spaces $\mathcal{M}_{0,n}$, equipped with a hidden holomorphic involution. This additional symmetry is associated to a split quaternionic structure. We introduce a categorical framework to present this object. Interaction between the geometry, physics and the arithmetics are discussed. An …

亏格3的曲线模空间中亏格105的一条完备曲线

A complete curve of genus 10 in the moduli space of curves of genus 3 and related questions

研究复数域上亏格3曲线模空间，通过特定方法证明其中存在一条亏格105的完备曲线。

Higher rank BN-theory for curves of genus 4

Higher rank Brill-Noether theory is completely known for curves of genus $\leq 3$. In this paper, we investigate the theory for curves of genus 4. Some of our results apply to curves of arbitrary genus.

The unirationality of the moduli space of curves of genus $\leq 14$

The new result is the unirationality of the moduli space of curves of genus 14. The method applies to lower genus.

Siegel modular forms and finite symplectic groups

The finite symplectic group Sp(2g) over the field of two elements has a natural representation on the vector space of Siegel modular forms of given weight for the principal congruence subgroup of level two. In this paper we decompose this representation, for various (small) values of the genus and the level, into irreducible representations. As a consequence we obtain uniqueness results for certain modular forms related to the superstring measure, a better understanding of certain modular forms in genus three studied by D'Hoker and Phong as well as a new construction of Miyawaki's cusp form of…

Log minimal model program for the moduli space of stable curves of genus three

In this paper, we completely work out the log minimal model program for the moduli space of stable curves of genus three. We employ a rational multiple $αδ$ of the divisor $δ$ of singular curves as the boundary divisor, construct the log canonical model for the pair $(\bar{\mathcal M}_3, αδ)$ using geometric invariant theory as we vary $α$ from one to zero, and give a modular interpretation of each log canonical model and the birational maps between them. By using the modular description, we are able to identify all but one log canonical models with existing compactifications of $M_3$, some ne…

The rationality of the moduli spaces of trigonal curves of odd genus

The moduli spaces of trigonal curves of odd genus $g>4$ are proven to be rational.

On hyperelliptic curves of genus 3

We study the moduli space of genus 3 hyperelliptic curves via the weighted projective space of binary octavics. This enables us to create a database of all genus 3 hyperelliptic curves defined over $\mathbb Q$, of weighted moduli height $\mathcal h =1$.

Cutting out arithmetic Teichmueller curves in genus two via Theta functions

We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants. The main technical tool is the computation of divisor classes of Hilbert Jacobi forms on the universal abelian surface over the pseudo-Hilbert modular surface.

Ekedahl-Oort strata on the moduli space of curves of genus four

We study the induced Ekedahl-Oort stratification on the moduli of curves of genus 4 in positive characteristic.

A canonical curve of genus 17

We compute equations for a Hurwitz curve of genus 17 and we conclude that the canonical ideal of any Hurwitz curve of genus 14 or 17 is generated by quadrics.

Families of genus two curves with many elliptic subcovers

We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to $\mathbb C$-isomorphism) defined over $\mathbb Q$, which…

$(p,d)$-elliptic curves of genus two

We study stable curves of arithmetic genus 2 which admit two morphisms of finite degree $p$, resp. $d$, onto smooth elliptic curves, with particular attention to the case $p$ prime.

On the rationality of the moduli of higher spin curves in low genus

The global geometry of the moduli spaces of higher spin curves and their birational classification is largely unknown for g >= 2 and r > 2. Using quite related geometric constructions, we almost complete the picture of the known results in genus g <= 4 showing the rationality of the moduli spaces of even and odd 4-spin curves of genus 3, of odd spin curves of genus 4 and of 3-spin curves of genus 4.

Admissible constants for genus 2 curves

S.-W. Zhang recently introduced a new adelic invariant for curves of genus at least 2 over number fields and function fields. We calculate this invariant when the genus is equal to 2.