Kimberly Hopkins, Jeffrey Stopple
We apply Tatuzawa's version of Siegel's theorem to derive two lower bounds on the size of the principal genus of positive definite binary quadratic forms.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The development of the princial genus theorem
本文简述了主种定理的发展，从高斯在二元二次型情形下提出的概念到埃米·诺特在类域理论中提出的主种定理的上同调表述。
Let $A$ be a definite quaternion algebra over $\mathbb Q$, with discriminant $D_A$, and $O$ a maximal order of $A$. We show that the minimum of the positive definite hamiltonian binary forms over $O$ with discrimiminant $-1$ is $\sqrt{D_A}$. When the different of $O$ is principal, we provide an explicit form representing this minimum, and when $O$ is principal, we give the list of the equivalence classes of all such forms. We also give criteria and algorithms to determine when the different of $O$ is principal.
An integral quadratic form is called strictly $n$-regular if it primitively represents all quadratic forms in $n$ variables that are primitively represented by its genus. For any $n \geq 2$, it will be shown that there are only finitely many similarity classes of positive definite strictly $n$-regular integral quadratic forms in $n + 4$ variables. This extends the recent finiteness results for strictly regular quaternary quadratic forms by Earnest-Kim-Meyer (2014).
Genus theta series, Hecke operators and the basis problem for Eisenstein series
给出了Hecke算子$T(p)$对正定积分二次型亏格θ级数作用的显式公式，并证明了亏格θ级数生成Eisenstein级数空间的一个定理.我们还讨论了我们的结果与Kudla的匹配原理θ积分的连接。
Let $\overlineρ: G_{\mathbf{Q}} \rightarrow {\rm GSp}_4(\mathbf{F}_3)$ be a continuous Galois representation with cyclotomic similitude character -- or, what turns out to be equivalent, the Galois representation associated to the $3$-torsion of a principally polarized abelian surface $A/\mathbf{Q}$. We prove that the moduli space $\mathcal{A}_2(\overlineρ)$ of principally polarized abelian surfaces $B/\mathbf{Q}$ admitting a symplectic isomorphism $B[3] \simeq \overlineρ$ of Galois representations is never rational over $\mathbf{Q}$ when $\overlineρ$ is surjective, even though it is both ratio…
In this paper, based on the theory of genus fields of biquadartic fields, we find a new larger family of biquadratic fields having a non-principal Euclidean ideal class, which implies the main results of \cite{JNT}.
The spinor type number formula for totally definite quaternion orders
本文研究了完全确定四元数代数的旋量类型数公式，证明了类型数可被相关商群阶整除，并扩展了类数可被类数整除的结果。
For a positive definite integral ternary quadratic form $f$, let $r(k,f)$ be the number of representations of an integer $k$ by $f$. The famous Minkowski-Siegel formula implies that if the class number of $f$ is one, then $r(k,f)$ can be written as a constant multiple of a product of local densities which are easily computable. In this article, we consider the case when the spinor genus of $f$ contains only one class. In this case the above also holds if $k$ is not contained in a set of finite number of square classes which are easily computable (see, for example, \cite{sp1} and \cite {sp2}). …
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…
Extending the notion of regularity introduced by Dickson in 1939, a positive definite ternary integral quadratic form is said to be spinor regular if it represents all the positive integers represented by its spinor genus (that is, all positive integers represented by any form in its spinor genus). Jagy conducted an extensive computer search for primitive ternary quadratic forms that are spinor regular, but not regular, resulting in a list of 29 such forms. In this paper, we will prove that there are no additional forms with this property.
The classification and representations of positive definite ternary quadratic forms of level 4N
本文针对无平方因子整数N，给出4N级正定三元二次型的完整分类，并利用三元二次型、四元数代数与Jacobi形式的关系，建立各类表示加权和的显式公式，进而得到类数公式及权3/2模形式Eisenstein空间的应用。
In this paper we first of all determine all possible genera of (odd and even) definite unimodular lattices over an imaginary-quadratic field. The main questions are whether the partial class numbers of lattices with given Steinitz class within one genus are equal for all occuring Steinitz classes and whether the partial masses of those partial genera are equal. We show that the answer to the first question in general is "no" by giving a counter example, while the answer to the second question is "yes" by proving a mass formula for partial masses. Finally, we determine a list of all single-clas…
We make some corrections to Nipp's tables of positive definite integral quaternary quadratic forms. They only affect $p$-adic densities and $p$-adic Jordan splittings in the Appendix of Nipp's book. There are no errors found in Nipp's list of quaternary forms for each genus.
On the nonexistence of certain curves of genus two
证明了：若q是奇素数的幂，则F_q上不存在亏格为2的曲线，其Jacobian的特征多项式为x^4 +（2- 2 q）x^2 + q^2.我们的证明利用了Q的双二次扩张中的Brauer关系，证明了F_q上的每个具有给定特征多项式的主极化阿贝尔曲面在F_{q^2}上分裂为极化椭圆曲线的乘积。
Let $p$ be an odd prime, $N$ a square-free odd positive integer prime to $p$, $π$ a $p$-ordinary cohomological irreducible cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbb{A}_\mathbb{Q})$ of principal level $N$ and Iwahori level at $p$. Using a $p$-integral version of Rallis inner product formula and modularity theorems for $\mathrm{GSp}_{4/\mathbb{Q}}$ and $\mathrm{U}_{4/\mathbb{Q}}$, we establish an identity between the $p$-part of the critical value at $1$ of the degree $5$ $L$-function of $π$ twisted by the non-trivial quadratic Dirichlet character $ξ$ associated to the exten…
We describe the intersection of the Torelli locus $j(\mathcal{M}_4^{ct}) = \mathcal{J}_4$ with Newton and Ekedahl-Oort strata related to the supersingular locus in characteristic two. We show that the locus of supersingular Jacobians $\mathcal{S}_4\cap\mathcal{J}_4$ in characteristic two is pure of dimension three. One way to obtain that result uses an analysis of the data of smooth genus four curves and principally polarized abelian fourfolds defined over $\mathbb{F}_2$, and another involves studying the Ekedahl-Oort types and using the indecomposability of some $\mathrm{BT}_1$ group schemes.
Genus two curves with quaternionic multiplication and modular jacobian
本文给出了一种确定带四元数乘法的模交换曲面$A_f$的所有主极化的同构类的方法，该模交换曲面带有一个不带复乘的规范化新形式$f$.我们包括一个例子，$A_f$与四元数乘法，我们发现数值曲线$C$的雅可比矩阵是$A_f$的数值逼近，我们证明了它有四元数乘法，是同源的$A_f$。
Constructing genus 2 curves with given refined Humbert invariants
本文提出算法，构造具有给定精细Humbert不变量的genus 2曲线
The classification of the refined Humbert invariant for curves of genus 2
本文研究双曲不变量的细化分类，当曲线的雅可比簇与具有复乘性的椭圆曲线积同构时，探讨其几何和算术性质。
Kneser's method of neighbors
本文综述了1957年Kneser提出的用于枚举正定整二次型亏格中类的邻域法，阐述了该方法的理论与实用价值及其在数论中的相关应用。
In this paper we study genus 2 curves whose Jacobians admit a polarized (4,4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion and we derive the relation their absolute invariants satisfy. As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus 2 curves, where previously only Richelot isogenies …
Hurwitz class number relations and mock modular forms
本文将Hurwitz经典类数关系推广，以正定二元二次型替代一元二次型，建立涉及Cohen广义类数的无穷族类数关系，并构造依赖亏格的降阶算子。
Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles appears to be closely related to an effective version of Faltings's theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of {\em Galois representations}, the structures linking motives to automorphic forms according to the Langl…
Bad Reduction of Genus Two Curves with complex multiplication
本文研究具有双二次CM域复乘的亏格二曲线，通过正交Shimura簇上的CM环显式描述其雅可比簇，推导稳定坏约化素数的上界并证明其存在性。
Binary quadratic forms and elliptic curves with analytic rank one
研究给定椭圆曲线及正定二元二次型，证明在相关二次型表示集合中存在无穷多\(d\)，使扭曲椭圆曲线\(dy^2 = f(x)\)解析秩为1。
Abelian surfaces of GL2-type as Jacobians of curves
本文研究了GL 2型阿贝尔簇的主极化同构类集。作为结果的应用，我们构造了\bar上的亏格为2的曲线C，C '/\Q，它们在\Q上是非同构的，而在\Q上共享同构的非极化模雅可比簇;我们还给出了一种方法，以获得\Q上亏格为2的曲线，其雅可比簇同构于二次\Q -曲线的Weil限制，并给出了例子.
Nontrivial elements of Sha explained through K3 surfaces
本文给出了亏格为2的曲线的雅可比矩阵的主齐性空间非平凡的一种新的证明方法。这个想法是展示一个Brauer-Manin障碍的存在有理点在这个主齐性空间的商。在一个显式的例子中，我们应用该方法表明，一个特定的曲线有无穷多个二次扭曲的雅可比矩阵有非平凡的Tate-Shafarevich群。
We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A_3 of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of d…
Siegel modular forms mod p
证明了特征p不等于2的域上亏格g>1的具有水平结构N的主极化阿贝尔簇模叠加的一个形式的消灭定理.这用于计算这些堆栈的Picard群。作为应用，我们得到了Siegel积分模形式之间的一个同余关系.
Siegel modular forms of genus 2 and level 2: cohomological computations and conjectures
本文研究了具有2级结构的主极化交换曲面的模空间上的某些局部系统的上同调。Frobenius在这些局部系统的平展上同调群的交错和上的迹可以通过在给定的有限域上计算亏格为2的点曲线的个数来计算。这种上同调与向量值Siegel模形式密切相关。在[FvdG]中进行了水平1中的相应方案。在这里，我们将其扩展到出现新现象的级别2。我们确定了Eisenstein上同调及其S_6作用量对全二阶结构的贡献，并在此基础上对内窥镜贡献作了精确的计算。我们还预测了齐藤-黑川升力的矢量值模拟的存在。假设这些假设是基于充分的数值证据，我们得到的Heck算子T（p）的p < 41的“真正的”西格尔模形式的剩余空间上的痕迹。我们提出了一些例子，1维空间的特征形，这些痕迹与赫克特征值相吻合。我们希望，专家在解除和内窥镜检查将能够证明我们的知识。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。