Still better nonlinear codes from modular curves
Noam D. Elkies
结合Goppa构造的两个新的改进，利用有限域k上具有多个有理点的模曲线的几何性质，给出了有限域k上非线性纠错码的一个新的构造.由此产生的代码是渐近目前已知的最好的。
We give a new construction of nonlinear error-correcting codes over suitable finite fields k from the geometry of modular curves with many rational points over k, combining two recent improvements on Goppa's construction. The resulting codes are asymptotically the best currently known.
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Remarks on codes from modular curves: MAGMA applications
讨论由曲线产生的AG或Goppa码的解释性论文，首先从抽象的一般角度，然后转向与模曲线相关的具体示例。我们将尝试用一个特殊的例子来解释这些非常技术性的思想，在一个典型的研究生的水平上，有一些背景的模形式，数论，群论，代数几何。包括许多使用MAGMA的例子。
Let q be a prime power and E a non-isotrivial elliptic curve over Fq(T) given by a Weierstrass model. We survey the construction, with an explicit point of view, of the modular parametrization of E by the associated Drinfeld modular curve. We then prove a formula that allows us to evaluate this modular parametrization at cusps and we produce an explicit method to compute these values. Finally we illustrate our results with several examples in characteristic 2 and 3.
Drinfeld modular curve and Weil pairing
本文给出了Drinfeld模曲线的紧化.这种紧化类似于Katz和Mazur给出的经典模曲线的紧化。我们展示了如何Weil配对的Drinfeld模块，我们在早期的工作中定义的Drinfeld模曲线上产生一个地图。我们介绍了泰特-德林费尔德模块，并展示了如何描述的形式邻域的计划的尖点的德林费尔德模曲线。
On hyperelliptic modular curves over function fields
证明了$\mathbb{F}_q(T)$上的$D$ -椭圆层的超椭圆的模曲线只有1000条.在奇特征中，我们给出了这类曲线的一个完全分类。
We determine the automorphism group of the modular curve $X_0^*(N)$, obtained as the quotient of the modular curve $X_0(N)$ by the group of its Atkin-Lehner involutions, for all square-free values of $N$.
Genus formula for modular curves of $D$-elliptic sheaves
证明了$D$ -椭圆层的模曲线的亏格公式。我们利用这个公式证明了当$D$的判别式的阶趋于无穷大时，$D$ -椭圆层的模曲线的约化达到Drinfeld-Vladut界.
On quadratic points of classical modular curves
经典模曲线是算术几何中的一个重要问题。在这项调查中，我们将展示如何Fumiyuki Momose的工作是为了列出经典的模曲线，满足二次点集在$\mathbb{Q}$是无限的。特别是我们回顾结果Momose超椭圆模曲线和自同构群的模曲线。此外，我们修正了现有文献中关于模曲线的自同构群的一些不准确之处，并且我们提供了难以找到精确参考的不同结果，例如：关于超椭圆曲线和双椭圆曲线的一个算术结果（就像哈里斯和西尔弗曼定理的算术陈述（或者阿布拉莫维奇和哈里斯定理的$d=2$））以及以X（N）为参数的$\mathbb{Q}$上椭圆曲线的导体上。
Finiteness results for modular curves of genus at least 2
有理数域Q上的曲线X是模曲线，如果对某个N，X被X_1（N）支配;另外，如果它在J_1（N）中的雅可比像包含在J_1（N）的新子簇中，则X称为新的模曲线。证明了对任意不小于2的整数g，亏格为g的Q上的新模曲线的集合是有限的且可计算的.对于可计算性的结果，我们证明了一个算法版本的de Franchis-Severi定理。对于有界性的新模曲线，对于雅可比矩阵是J_0（N）的新部分与N可被指定素数整除的商的新模曲线，以及对于水平在限制集中的模曲线（新的或非新的），证明了类似的有限性结果.我们详细研究了新的模超椭圆曲线。特别是，我们发现所有新的模曲线的亏格2明确，并构建可能是所有新的模超椭圆曲线的所有属的完整列表。最后证明了对特征为零的域k和至少为2的整数g，k上由Fermat曲线支配的亏格g曲线集是有限的且可计算的。
In this paper we prove that there are finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we show that none can admit a smooth plane model of degree 5, 6 or 7. Further, if a modular curve of Shimura type admits a smooth plane model of degree 8 we show that it must be a twist of one of four curves.
Higher Hida theory for Drinfeld modular curves
受Boxer和Pilloni启发，为Drinfeld模曲线上的线丛上同调建立高阶Hida理论，并插值Serre对偶。
We describe the ring of modular forms of degree 2 in characteristic 2 using its relation with curves of genus 2.
Automorphisms of hyperelliptic modular curves $X_0(N)$ in positive characteristic
研究素数$ p\nmid N$上模曲线$X_0(N)$的约化$X_0(N) \times \bar{\mathbb{F}}_p$的自同构群.
CM points on products of Drinfeld modular curves
设X是奇特征一般基环A上Drinfeld模曲线的乘积.我们对X的包含CM点的Zurkiki稠密集的那些子簇进行分类。这是安德烈-奥尔特猜想的一个类比。作为应用，我们构造了整体函数域上模椭圆曲线上的非平凡高Heegner点族.
This paper presents the theory of holomorphic vector valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of SL_2(Z) play in the holomorphic theory of vector valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other aut…
The Langlands-Kottwitz approach for the modular curve
我们展示了如何Langlands-Kottwitz方法可以用来确定局部因子的Hasse-Weil zeta函数的模曲线的地方坏约化。在此基础上，我们证明了Haines和Kottwitz的一个猜想。
Modular Curves, Modular Surfaces, and Modular Fourfolds
本文综述了作者最近关于Hilbert模四重性的一些工作。在讨论了交换域上X的上同调和特殊余维2的Hirzebruch-Zagier型圈Z之后，给出了交换域上X的Tate猜想的一个证明.接着描述了一种在Bloch Chow群CH ^3（X，1）中构造类的方法，该方法通过使用如上所述的圈Z的Hecke平移，并在模曲线的平移中具有适当的交点。本文最后引入了一般Shimura簇X的模Gersten复形及其相应的群CH^ p_{mod}（X）和CH^ p_{mod}（X，1）.
Explicit towers of Drinfeld modular curves
我们给出了任何有限域上Drinfeld模曲线的最简单塔的显式方程，并观察到它们与Garcia和Stichtenoth构造的曲线的渐近最优塔一致。
We determine which of the modular curves $X_Δ(N)$, that is, curves lying between $X_0(N)$ and $X_1(N)$, are bielliptic. Somewhat surprisingly, we find that one of these curves has exceptional automorphisms. Finally we find all $X_Δ(N)$ that have infinitely many quadratic points over $\mathbb{Q}$.
Modularity of some elliptic curves over totally real fields
研究了一般全实数域上椭圆曲线的模性，建立了非模$j$ -不变量集的有限性结果.通过分析某些模曲线上的二次点，证明了某些实二次域上的椭圆曲线都是模曲线。
The modular automorphisms of quotient modular curves
本文确定了 4、9 不整除层数 N 时任意商模曲线的模自同构群，发现涉及 w_25 时的非预期 3 阶自同构，并推出充分大 N 下 X_0^*(N^2) 的完整自同构群。
We classify the three-dimensional representations of the modular group that are reducible but indecomposable, and their associated spaces of holomorphic vector-valued modular forms. We then demonstrate how such representations may be employed to compute periods of modular curves. This technique obviates the use of Hecke operators, and therefore provides a method for studying noncongruence modular curves as well as congruence.
The $E_8$ root lattice can be constructed from the modular curve $X(13)$ by the invariant theory for the simple group $\text{PSL}(2, 13)$. This gives a different construction of the $E_8$ root lattice. It also gives an explicit construction of the modular curve $X(13)$.
Elliptic functions and equations of modular curves
设$p\ge 5$为素数。我们发现，空间的重量一爱森斯坦系列定义嵌入到$\PP^{(p-3)/2}$的模曲线$X_1(p)$的同余组$Γ_1(p)$，这是计划理论上切出的显式二次方程。
In this paper, we introduce modular polynomials for the congruence subgroup $Γ_0(M)$ when $ X_0(M) $ has genus zero and therefore the polynomials are defined by a Hauptmodul of $ X_0(M) $. We show that the intersection number of two curves defined by two modular polynomials can be expressed as the sum of the numbers of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of positive definite binary quadratic forms over $\mathbb{Z}$. We also show that the intersection numbers can be also combinatorially written by Fourier coefficients of the Siegel Eisenstein series of degree 2, weight 2 with respec…
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。
Torsion points on modular curves and Galois theory
给出了M的一个定理的证明. Baker和A. Tamagawa提出了一种新的方法，利用几乎有理扭点的概念，给出了嵌入其雅可比矩阵中的模曲线上的扭点的完整描述。'
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 $X_0^+(N…
Bielliptic and Hyperelliptic modular curves X(N) and the group Aut(X(N))
我们确定了所有超椭圆或双椭圆的模曲线X（N）（用$N\geq 7$）.本文证明了X（N）的自同构群与$Γ(N)$在$\operatorname{PSL}_2(\mathbb{R})$中的正规化子重合.
Weber modular curves and modular isogenies
本文研究韦伯函数定义的模曲线及其关联的模多项式、SL₂(ℤ)作用及椭圆曲线参数化，旨在探讨其确定的同源图，特别是奇异椭圆曲线的性质，并应用于密码学中的高效同源计算及显式伽罗瓦表示。
Abelian etale coverings of modular curves over local fields
我们涉及到的一部分交换etale基本群曲线的局部域的雅可比矩阵的Neron模型的组件组。我们把这个结果应用到模曲线X_0（p）/Q_p上，证明了X_0（p）上的非分歧阿贝尔覆盖X_1（p）（Shimura覆盖）用尽了X_0（p）的特殊纤维上的所有可能分歧。
Real components of modular curves
研究了模曲线的实分量。我们的主要结果是一个抽象的群论描述的实分量的模曲线所定义的一个同余子群的水平N的相应的子群SL 2（Z/NZ）。我们将这一结果应用于几类模曲线（如X_0（N），X_1（N）等）.得到实数分量的个数的公式。有些令人惊讶的是，2模N的乘法阶在许多情况下有很强的影响：例如，如果N是奇素数，则X_1（N）的实轨迹连通当且仅当-1和2生成（Z/NZ）^*。
In this study, we determine all modular curves $X_0(N)$ that admit infinitely many cubic points.
Gonality of modular curves in characteristic p
设k是特征为p的代数闭域，X（p^e;N）是参数化具有全层N结构（其中p不整除N）和全层p^e Igusa结构的椭圆曲线的曲线.所谓模曲线，我们指的是任何X（p^e;N）与（（Z/p^e Z）^* x \SL_2（Z/NZ））/{+-1}的任何子群的商。我们证明了在任何序列的不同模曲线在k上，k-共度趋于无穷大。这扩展了早期的工作，其中该结果针对特定的模曲线序列得到了证明，例如对于p不整除N的X_0（N）。给出了Galois像在椭圆曲线挠上的一致有界性陈述在函数场模拟中的一个应用。
Modular congruences, Q-curves, and the diophantine equation x^4 + y^4 = z^p
本文证明了关于广义Fermat方程$x^4+y^4=z^p$的两个结果。特别地，我们证明了第一种情况是真的，如果$p \neq 7$。
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。
Effective Siegel's theorem for modular curves
我们证明了在除1/2个模曲线外的所有模曲线上，以及在除1个素幂水平的模曲线外的所有模曲线上，都可以有效地确定整点。
We determine the automorphism group of the split Cartan modular curves $X_{\operatorname{split}}(p)$ for all primes $p$.
Stable reduction of modular curves
我们确定了伽罗瓦群${\rm SL}_2(p)$的射影直线的所有三个点覆盖在$p$上的稳定约化。作为一个特例，我们恢复了Deligne和Rapoport关于模曲线的约化的结果：$X_0(p)$和$X_1(p)$ .我们的方法没有利用模曲线是模空间这一事实。相反，我们依赖于雷诺和作者的结果描述了稳定的减少三个点覆盖的伽罗瓦群是严格可分的$p$。
Runge's Method and Modular Curves
在满足一定的Runge条件的前提下，利用定义曲线的同余群，给出了任意域上任意模曲线上S-整点的j-不变量的界.然后，我们应用我们的边界证明，对于足够大的素数p，$X_0^+ (p^r)(Q)$的点与r>1的尖点或CM点。这可以解释为不存在具有较高素幂次数的二次椭圆Q曲线。
Among all the dynamical modular curves associated to quadratic polynomial maps, we determine which curves have infinitely many quadratic points. This yields a classification statement on preperiodic points for quadratic polynomials over quadratic fields, extending previous work of Poonen, Faber, and the authors.
K1 of products of Drinfeld Modular Curves and Special Values of L-functions
设$X_0(I)$是具有水平$I$结构的Drinfeld模曲线，其中$I$是$\F_{q}[T]$中的一元无平方理想.本文证明了在原动机上同调群$H^3_{\M}(X_0(I) \times X_0(I),\Q(2))$中存在一个元素，其调节子与Ranking-Selberg卷积$L$ -函数的一个特殊值有关.这个结果是模曲线的自积的Beilinson定理的函数域模拟。
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 $N$.
Let $F_n$ denote the Fermat curve given by $x^n+y^n=z^n$ and let $μ_n$ denote the Galois module of $n$th roots of unity. It is known that the integral homology group $H_1(F_n,\Z)$ is a cyclic $\Z[μ_n\times μ_n]$ module. In this paper, we prove this result using modular symbols and the modular description of Fermat curves; moreover we find a basis for the integral homology group $H_1(F_n,\Z)$. We also construct a family of Fermat curves using the Fermat surface and compute its monodromy.
We show that if p is a prime, then all elliptic curves defined over the cyclotomic Z_p extension of Q are modular.
Quadratic modular symbols on Shimura curves
引入了二次模符号的概念，并研究了这些符号与二次p进L函数的关系。这些对象在[3]中在模曲线的情况下引入。本文讨论了一种将二次模符号和二次p-adic L-函数附加到更一般的Shimura曲线上的方法。
We describe an algorithm for computing a $\Q$-rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining $q$-expansions for a basis of the corresponding space of cusp forms. We also give a moduli interpretation for general morphisms between modular curves.
Computing modular equations for Shimura curves
在经典设置中，模曲线$X_0(1)$的水平$N$的模方程是由$j(τ)$和$j(Nτ)$满足的多项式关系，其中$j(τ)$是标准椭圆$j$ -函数。在本文中，我们将描述一种方法来计算模块方程的设置志村曲线。主要内容是文[13]中提出的计算Shimura曲线上模形式空间上Hecke算子的显式方法。
Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy $\text{PGL}_2(\mathbb{F}_q)$ or $\text{PSL}_2(\mathbb{F}_q)$. We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus.
Local diophantine properties of modular curves of $\cal{D}$-elliptic sheaves
研究了局部域上$\cal{D}$ -椭圆层的模曲线上有理点的存在性及其特殊纤维的结构.讨论了它的一些应用，包括寻找由四元数代数产生的算术群的表示，寻找$\cal{D}$ -椭圆层的模曲线的方程，以及构造违反Hasse原理的曲线.
In this paper, we propose an improved algorithm for computing mod $\ell$ Galois representations associated to eigenforms of arbitrary levels prime to $\ell$. Precisely, we present a method to find the Jacobians of modular curves which have the smallest possible dimensions in a well-defined sense to realize the modular Galois representations. This algorithm also works without the assumption $\ell \ge k-1$, where $k$ are the weights of the eigenforms.
We prove that every elliptic curve defined over a totally real number field of degree 4 not containing $\sqrt{5}$ is modular. To this end, we study the quartic points on four modular curves.
Cubic points on dynamical modular curves
本文针对与二次多项式映射关联的动力模曲线族，判定含无穷多三次点的曲线类型，据此给出三次域上二次多项式预周期点的分类结论，拓展了相关已有研究成果。
We consider the generalized Jacobian $\widetilde{J}_0(N)$ of a modular curve $X_0(N)$ with respect to a reduced divisor given by the sum of all cusps on it. When $N$ is a power of a prime $\geq 5$, we exhibit that the group of rational torsion points $\widetilde{J}_0(N)(\mathbb{Q})_{\mathrm{Tor}}$ tends to be much smaller than the classical Jacobian.
On the arithmetic of certain modular curves
在这项工作中，我们估计的$X_1(N)$和$X_0(N),$之间的中间模曲线的亏格，并确定所有的三角形。
We present a detailed analysis of how to implement the computation of modular symbols for a given elliptic curve by using numerical approximations. This method turns out to be more efficient than current implementation as the conductor of the curve increases.
Supersingular reduction and strongly special intersections in powers of the modular curve
本文基于Daworrpap和Dawpap的工作，在假设弱化Lang-Trotter猜想下，证明了Y(1)^n中的两个Zilber-Pink型陈述。
Approximate computations with modular curves
本文介绍了算法数论的一些最新进展，着重介绍了用模曲线及其雅可比行列式进行近似计算的关键作用，以供对代数几何和数论中的数值计算感兴趣的数学家参考。这些近似值在维数和所需的有效数字数方面在多项式时间内完成。我们解释的主要思想是如何做的近似，说明他们的例子，我们勾勒出一些应用在数论。
Elliptic Curves over Real Quadratic Fields are Modular
证明了定义在实二次域上的所有椭圆曲线都是模曲线。
Bounding the $j$-invariant of integral points on certain modular curves
本文给出了一类具有正亏格且尖点小于三个的模曲线上整点的$j$ -不变量的两个有效界。
A Modular Curve of Level 9 and the Class Number One Problem
本文给出了9阶非分裂Cartan子群的正规化子所对应的模曲线的一个显式参数化。我们确定了这条模曲线的所有积分点。作为应用，我们给出了类数为1的问题的另一种解法。
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