The development of the princial genus theorem
Franz Lemmermeyer
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。
In this article we sketch the development of the principal genus theorem from its conception by Gauss in the case of binary quadratic forms to the cohomological formulation of the principal genus theorem of class field theory by Emmy Noether.
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
In this paper, we remove certain hypothesis in the theorem of Bertolini-Darmon on the rationality of Stark-Heegner points over narrow genus class fields of real quadratic fields. Along the way, we establish that certain normalized special values of $L$-functions are squares of rational numbers, a result that is of independent interest, and can be regarded as instances of the rank zero case of the Birch and Swinnerton-dyer conjecture modulo squares.
Lower bounds for the principal genus of definite binary quadratic forms
我们应用Tatuzawa的Siegel定理的版本来获得两个下界的大小的主种的正定二元二次型。
Rudnick and Soundararajan's Theorem for Function Fields
本文证明了Rudnick和Soundararajan关于二次Dirichlet $L$ -函数矩下界的一个定理的函数域形式。我们建立了固定有限域$\mathbb{F}_{q}$上亏格为$g$的超椭圆曲线的二次Dirichlet $L$-函数在大亏格$g$极限下的矩的下界.
Genus fields of abelian extensions of congruence rational function fields
在这篇论文的公开版本[Finite Fields and Their Applications {\bf 20}（2013）40--54]中，在该论文定理4.2的证明中存在错误。本文对定理4.2，4.5和5.2给出了同余函数域的亏格域的构造。首先，我们考虑割圆函数域的情况下的想法Leopoldt，然后一般情况下。作为应用，我们给出了Kummer，Artin-Schreier和循环$p$-扩张的亏格域. Kummer扩张是由G. Peng和Artin-Schreier扩张由S. Hu和Y.李
We construct curves of each genus $g\geq 2$ for which Coleman's effective Chabauty bound is sharp and Coleman's theorem can be applied to determine rational points if the rank condition is satisfied. We give numerous examples of genus two and rank one curves for which Coleman's bound is sharp. Based on one of those curves, we construct an example of a curve of genus five whose rational points are determined using the descent method together with Coleman's theorem.
A central limit theorem for prime geodesics on random surfaces of large genus
该研究证明了大亏格Weil–Petersson随机闭双曲曲面上，范数在特定区间内的素测地线的归一化加权计数满足渐近高斯分布，拓展了素测地线计数的相关结论。
In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the value of these modular functions at CM points of the Siegel upper-half space, which, if their…
Siegel's Theorem and the Shafarevich Conjecture
已知在超椭圆曲线的情况下，可以使Shafarevich猜想有效，即，对任意数域k和k的任意有限集S，可以有效地计算出k上超椭圆曲线在S外具有好约化的同构类集.我们证明了这个结果的推广到亏格为g的超椭圆曲线的Jacobian的有效Shafarevich猜想将意味着亏格为g的超椭圆曲线上整点的Siegel定理的有效版本。
We study the fluctuations of smooth linear statistics of Laplace eigenvalues of compact hyperbolic surfaces lying in short energy windows, when averaged over the moduli space of surfaces of a given genus. The average is taken with respect to the Weil-Petersson measure. We show that first taking the large genus limit, then a short window limit, the distribution tends to a Gaussian. The variance was recently shown to be given by the corresponding quantity for the Gaussian Orthogonal Ensemble (GOE), and the Gaussian fluctuations are also consistent with those in Random Matrix Theory, as conjectur…
Computing nonsurjective primes associated to Galois representations of genus $2$ curves
本文针对雅可比簇仅有平凡几何自同态的亏格2曲线，提出一种实用算法，用于计算使伽罗瓦挠点作用不极大的有限素数集，并已在LMFDB全部相关曲线上完成验证。
A Note on Pépin's counter examples to the Hasse principle for curves of genus 1
在C.R.发表的一系列文章中，一个多世纪前的巴黎，T。佩潘宣布了一系列关于$ax^4 + by^4 = z^2$型丢番图方程可解性的“定理”。在这篇文章中，我们展示了如何证明这些索赔使用的结构2类群的虚二次数域。我们还将从现代的角度来研究一些相关的结果（包括$n = 7$的FLT）。
The Chabauty-Coleman method is a $p$-adic method for finding all rational points on curves of genus $g$ whose Jacobians have Mordell-Weil rank $r < g$. Recently, Edixhoven and Lido developed a geometric quadratic Chabauty method that was adapted by Spelier to cover the case of geometric linear Chabauty. We compare the geometric linear Chabauty method and the Chabauty-Coleman method and show that geometric linear Chabauty can outperform Chabauty-Coleman in certain cases. However, as Chabauty-Coleman remains more practical for general computations, we discuss how to strengthen Chabauty-Coleman t…
In this note, an analogous statement to the Nagell-Lutz theorem does not hold for the Jacobian of a certain curve of genus 2 over $\mathbb{C}(t)$. As a by-product, we give a (2, 3, 6) quasi-torus decomposition for the dual curve of a smooth cubic.
We discuss a multiplicative counterpart of Freiman's $3k-4$ theorem in the context of a function field $F$ over an algebraically closed field $K$. Such a theorem would give a precise description of subspaces $S$, such that the space $S^2$ spanned by products of elements of $S$ satisfies $\dim S^2 \leq 3 \dim S-4$. We make a step in this direction by giving a complete characterisation of spaces $S$ such that $\dim S^2 = 2 \dim S$. We show that, up to multiplication by a constant field element, such a space $S$ is included in a function field of genus $0$ or $1$. In particular if the genus is $1…
We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlevé irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem which have interesting consequences…
Prime geodesic theorem and closed geodesics for large genus
本文研究大亏格双曲曲面模空间中素数测地线定理的误差上界及其期望值，并利用所得结果证明长度显著小于√g的闭测地线大多简单且不可分离、长度显著大于√g的闭测地线大多非简单，从而证实Lipnowski-Wright猜想。
We study the $p$-rank stratification of the moduli space of Prym varieties in characteristic $p > 0$. For arbitrary primes $p$ and $\ell$ with $\ell \not = p$ and integers $g \geq 3$ and $0 \leq f \leq g$, the first theorem generalizes a result of Nakajima by proving that the Prym varieties of all the unramified ${\mathbb Z}/\ell$-covers of a generic curve $X$ of genus $g$ and $p$-rank $f$ are ordinary. Furthermore, when $p \geq 5$ and $\ell = 2$, the second theorem implies that there exists a curve of genus $g$ and $p$-rank $f$ having an unramified double cover whose Prym has $p$-rank $f'$ fo…
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。