Jing Long Hoelscher
This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur as such Galois groups. Also the explicit structure of ray class groups of regular cyclotomic number field is given.
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
We classify all cyclotomic matrices over real quadratic integer rings and we show that this classification is the same as classifying cyclotomic matrices over the compositum all real quadratic integer rings. Moreover, we enumerate a related class of symmetric matrices; those matrices whose eigenvalues are contained inside the interval [-2,2] but whose characteristic polynomials are not in Z[x].
We prove that all elliptic curves defined over the cyclotomic $\mathbb{Z}_p$-extension of a real quadratic field are modular under the assumption that the algebraic part of the central value of a twisted $L$-function is a $p$-adic unit. Our result is a generalization of a result of X. Zhang, which is a real quadratic analogue of a result of Thorne.
In this paper we study the possible torsions of elliptic curves over $\mathbb Q(i)$ and $\mathbb Q(\sqrt {-3})$.
A note on quadratic cyclotomic extensions
本文通过引入关键映射并确定特定素数幂下的最大自然数，给出了任意域上二次分圆扩张中本原单位根的两种统一刻画。
Let $\mathcal{A}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of a real quadratic number field $F$. The cardinality of $\mathcal{A}_n$ is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the $\mathcal{A}_n$'s stabilizes for the real fields $F=\mathbb{Q}(\sqrt{f})$ for any integer $0<f<10000$. Equivalently Greenberg's conjecture holds for those fields.
Let $K = \mathbb{Q}(\sqrt{-3})$ or $\mathbb{Q}(\sqrt{-1})$ and let $C_n$ denote the cyclic group of order $n$. We study how the torsion part of an elliptic curve over $K$ grows in a quadratic extension of $K$. In the case $E(K)[2] \approx C_2 \oplus C_2$ we determine how a given torsion structure can grow in a quadratic extension and the maximum number of quadratic extensions in which it grows. We also classify the torsion structures which occur as the quadratic twist of a given torsion structure.
On CM elliptic curves and the cyclotomic $λ$-invariants of imaginary quadratic fields
本文研究虚二次域分圆 $\mathbb{Z}_p$-扩张的 Iwasawa $\lambda$-不变量，证明其大于 $1$ 等价于特定 CM 椭圆曲线约化后的点数被 $p^2$ 整除。
The Parity of Two Types of Cyclotomic Euler Sums via Contour Integrals
本文利用围道积分与留数方法研究两类分圆欧拉型和的奇偶性，给出线性与二次情形的显式公式，并推广到任意阶及更一般条件。
We determine all Hermitian $\mathcal{O}_{\Q(\sqrt{d})}$-matrices for which every eigenvalue is in the interval [-2,2], for each d in {-2,-7,-11,-15\}. To do so, we generalise charged signed graphs to $\mathcal{L}$-graphs for appropriate finite sets $\mathcal{L}$, and classify all $\mathcal{L}$-graphs satisfying the same eigenvalue constraints. We find that, as in the integer case, any such matrix / graph is contained in a maximal example with all eigenvalues $\pm2$.
Let $K = \mathbb{Q}(\sqrt{-3})$ or $\mathbb{Q}(\sqrt{-1})$ and let $C_n$ denote the cyclic group of order $n$. We study how the torsion part of an elliptic curve over $K$ grows in a quadratic extension of $K$. In the case $E(K)[2] \approx C_1$ we investigate how a given torsion structure can grow in a quadratic extension and the maximum number of extensions in which it grows. We also study the torsion structures which occur as the quadratic twist of a given torsion structure. In order to achieve this we examine $N$-isogenies defined over $K$ for $N=15,20,21,24,27,30,35$.
Iwasawa module of the cyclotomic $\mathbb{Z}_{2}$-extension of certain real quadratic fields
本文针对实二次域的割圆$\mathbb{Z}_{2}$-扩张，在岩泽模循环的情形下构造了Greenberg猜想成立的新例子，并利用双二次域的基本单位系给出了计算岩泽模阶数的方法。
On the modularity of elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension of some real quadratic fields
本文遵循Thorne的思想，利用自守性提升定理与Iwasawa理论，在给定假设下证明了某些实二次域的$\mathbb{Z}_p$-扩张上椭圆曲线的模性。
In a previous paper, the author examined the possible torsions of an elliptic curve over the quadratic fields $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$. Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$, we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves o…
We formulate and for the most part prove a conjecture in the style of Mazur-Greenberg for the nonvanishing of central values of Rankin-Selberg $L$-functions attached to elliptic curves in abelian extensions of imaginary quadratic fields. This in particular generalizes the theorem of Rohrlich on $L$-functions of elliptic curves in cyclotomic towers to the setting of abelian extensions of imaginary quadratic fields, corresponding to families of degree-four $L$-functions given by $\operatorname{GL}(2)\times\operatorname{GL}(2)$ Rankin-Selberg $L$-functions. It also generalizes the theorems of Roh…
In this paper we establish a connection between the Gauss factorials and Iwasawa's cyclotomic $λ$-invariant for an imaginary quadratic field $K$. As a result, we will explain a corespondance between the 1-exceptional primes of Cosgrave and Dilcher for $m = 3$ and $m = 4$, and the primes for which the $λ$-invariants for $K = \mathbb{Q}(\sqrt{-3})$ and $K = \mathbb{Q}(i)$ is greater than one, respectively. We refer to the latter primes as ``non-trivial'' for their respective fields. We will also see that similar correspondences are true for $K = \mathbb{Q}(\sqrt{-d})$ when $d = 2,5$ and $6$. As …
Kitaoka's conjecture for real cyclotomic fields
该研究证明了所有实分圆域上的Kitaoka猜想，明确这类域存在通用经典三元二次型当且仅当它是实二次域。
Let K denote the quadratic field $\mathbb{Q}(\sqrt{d})$ where d=$-1$ or $-3$. Let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian $2$-extension of $K$.
In this paper, by using the theory of circulant matrices we study some matrices over finite fields which involve the quadratic character and trinomial coefficients.
Heuristics for anti-cyclotomic $\mathbb{Z}_p$-extensions
本文提出交集启发式与不变量启发式，并结合计算研究反分圆 $\mathbb{Z}_p$-塔中的岩泽不变量，预测类域相交规律及 $λ$、$μ$ 在非分裂情形下通常消失。
Cotorsion of anti-cyclotomic Selmer groups on average
本文研究椭圆曲线在改变素数p或二次虚域时，Selmer群在反分圆Z_p-扩张上呈现余挠性质的平均情况。
$p$-class groups in the cyclotomic $\mathbb{Z}_p$-extension of imaginary quadratic fields
本文研究虚二次域分圆$\mathbb{Z}_p$-扩张中$p$不分裂时第一层$p$-类群的结构，得到循环因子增长约束，并证明许多情况下Iwasawa $\lambda$-不变量由第一层$p$-秩决定，最后用Cohen-Lenstra-Martinet哲学分析其概率分布。
The absolute Galois group of the cyclotomic field $K={\mathbb Q}(ζ_p)$ acts on the étale homology of the Fermat curve $X$ of exponent $p$. We study a Galois cohomology group which is valuable for measuring an obstruction for $K$-rational points on $X$. We analyze a $2$-nilpotent extension of $K$ which contains the information needed for measuring this obstruction. We determine a large subquotient of this Galois cohomology group which arises from Heisenberg extensions of $K$. For $p=3$, we perform a Magma computation with ray class fields, group cohomology, and Galois cohomology, which determin…
Cyclotomic matrices and hypergeometric functions over finite fields
本文借助有限域上的超几何函数，研究了涉及特征与二元二次型的分圆矩阵的算术性质，并证实了Zhi-Wei Sun提出的相关猜想。
For a real quadratic field $K= \mathbb{Q}(\sqrt{d})$ with discriminant $D_{K}$ having four distinct prime factors, we study the structure of the $2$-class group $A(K_{1})$ of the first layer $K_{1} = \mathbb{Q}(\sqrt{2},\sqrt{d})$ of the cyclotomic $\mathbb{Z}_{2}$-extension of $K$. With some suitably convenient assumptions on the rank and the order of $A(K_{1})$, we characterize $K$ for which the $2$-class group $A(K)$ is isomorphic to $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$. We infer that the $2$-ranks of the class groups in each layer stabilizes by virtue of a result of Fukud…
Let $H= \mathbb{Q}(ζ_{n} + {ζ_{n}}^{-1})$ and $\ell$ be an odd prime such that $q \equiv 1 \pmod \ell$ for some prime factor $q$ of $n$. We get a bound on the $\ell$-rank of the class group of $H$(under some conditions) in terms of the $\ell$-rank of the class group of real quadratic subfield contained in $H$. This is an extension of a recent work of E. Agathocleous (with alternate hypothesis) where she handles $\ell=3$ case. As an application of our main result we relate the $\ell$-rank of real quadratic subfields of $H$.
A lower bound for the Call-Silverman height in cyclotomic extensions
该研究针对数域上的二次后临界有限多项式，结合位势理论等技术，建立分圆扩张中游荡代数整数的Call-Silverman高度下界，并给出非整数情形的部分结果。
Cyclotomic function fields over finite fields with irreducible quadratic modulus
本文研究有限域上首一二次不可约模的分圆函数域，给出其刻画并确定奇特征下的全自同构群，推广了已有相关结果。
On the Iwasawa $λ$-invariant of the cyclotomic $\mathbb{Z}_2$-extension of a family of real quadratic fields in which $2$ splits
本文研究实二次域的环形$\mathbb{Z}_2$-扩张的Greenberg猜想，证明在特定条件下$\lambda_2(K)=0$，方法结合Greenberg准则和Kumakawa的capitulation论证。
Given a number field $K$ and a polynomial $f(z) \in K[z]$ of degree at least 2, one can construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points for $f$, with an edge $α\to β$ if and only if $f(α) = β$. Restricting to quadratic polynomials, the dynamical uniform boundedness conjecture of Morton and Silverman suggests that for a given number field $K$, there should only be finitely many isomorphism classes of directed graphs that arise in this way. Poonen has given a conjecturally complete classification of all such directed graphs over $\mathbb{Q}$, whi…
Following the prequel work \cite{VO3}, we prove a generalization of "Mazur's conjecture" for $L$-functions of elliptic curves in abelian extensions of imaginary quadratic fields, including the assertion that the Mordell-Weil rank of an elliptic curve in the ${\bf{Z}}_p^2$-extension is finitely generated modulo Heegner points. The novelty of the approach here is to use the existence of a suitable $p$-adic $L$-function to reduce the problem to a minimal nonvanishing criterion, which should be applicable to a broader class of problems than considered here.
Let p be an odd prime number. In this paper, we show existence of certain infinite families of imaginary quadratic fields in which p splits and whose Iwasawa λ-invariant of the cyclotomic Zp-extension is equal to 1.
We construct certain elements in the integral motivic cohomology group $H^3_{\cal M}(E \times E',\Q(2))_{\ZZ}$, where $E$ and $E'$ are elliptic curves over $\Q$. When $E$ is not isogenous to $E'$ these elements are analogous to `cyclotomic units' in real quadratic fields as they come from modular parametrisations of the elliptic curves. We then find an analogue of the class number formula for real quadratic fields. Finally we use the Beilinson conjectures for $E \times E'$ to deduce them for products of $n$ elliptic curves. A certain amount of this paper is expository in nature.
Even better sums of squares over quintic and cyclotomic fields
分类了所有次数不超过5且存在有理整数系数万有二次型的全实域，解决了次数≤5的万有型提升问题，主要工具是每个2的倍数都是平方和的域的密集计算分类，并推广到高次实分圆域证明了Kitaoka猜想。
A Note on Cyclotomic Function Fields with Quadratic Modulus
本文研究了具有二次模的 cyclotomic 函数域的特征，证明了其与特定函数域的等价条件，包括群结构、曲率和有理地方数。
Unit Reducible Cyclotomic Fields
本文研究分圆域的单位可约化性，确定了若干小导子分圆域的单位可约化性，证明了单位可约化分圆域仅有有限多个，并给出了全正元素相关迹的渐近性质。
We show that several orbital equations and orbital clusters of the quadratic (logistic) map coincide surprisingly with cyclotomic {\it period equations}, polynomials whose roots are Gaussian periods. An analytical expression for the field discriminant of period equations is obtained and applied to discover and to fill gaps in number field databases constructed by numerical search processes. Such expression allows easy assess to inessential divisors of conventional discriminants and sheds light into why numerical construction of databases is a hard problem. It also provides significant informat…
In this paper we determine sufficient conditions for a quaternion algebra to split over a quadratic field. In the last section of the paper, we find a class of division symbol algebras of degree $n$ (where $n$ is a positive integer, $n\geq 3$) over a $p-$ adic field or over a cyclotomic field.
Nonvanishing of $L$-function of some Hecke characters on cyclotomic fields
本文证明某些 Hecke 特征在分圆域上的非零性，通过计算特征函数和 Weil 表示作用，应用于 Fermat 曲线和超椭圆曲线的 Jacobian，得到无穷多个解析秩为零的 twist。
The local structure of refined class groups in cyclotomic Iwasawa theory
该研究给出分圆伊瓦萨理论中射线类精细结构的显式局部描述，证明了特定条件下 $\mathbb{Q}(\zeta_p)$ 的 $p$ 主射线类群分量非零的等价条件，导出伊瓦萨主猜想的射线类形式，恢复 Finat 公式并推导出精细 $\mu$ 消失结果。
Cyclotomic and abelian points in backward orbits of rational functions
本文研究数域上有理函数向后轨道中的阿贝尔点与分圆点，证明阿贝尔点生成扩张仅在有限素数分歧，并给出二次有理函数及Lattès映射的相关分类与条件。
In this article we prove that (1-zeta+zeta^2) is a unit in the ring of integers of the cyclotomic field where zeta is a primitive n-th root of unity and n is coprime to 2 and 3. We also prove that for prime n, N_{Q(zeta)/Q}(1-zeta-zeta^2)=L(p) the p-th Lucas number thus completing the study of norms of quadratic polynomials in zeta that only have coefficients equal to 1 or -1 and both numbers appear.
Greenberg's conjecture for real quadratic number fields
本文计算了判别式小于100,000的实二次域的分圆Z_3-扩张中的3-类群，发现其阶在无穷时保持有界，从而验证了Greenberg猜想。
On fine Mordell-Weil groups over $\mathbb{Z}_p$-extensions of an imaginary quadratic field
本文研究虚二次域上椭圆曲线的精细Mordell-Weil群，在复乘、好普通约化及广义Heegner假设下推广Lei的结果，并建立BDP-Selmer群与p-adic L-函数和秩增长的关系。
Let $k_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of an algebraic number field $k$. We denote by $S$ a finite set of prime numbers which does not contain $p$, and $S(k_\infty)$ the set of primes of $k_\infty$ lying above $S$. In the present paper, we will study the structure of the Galois group $\mathcal{X}_S (k_\infty)$ of the maximal pro-$p$ extension unramified outside $S (k_\infty)$ over $k_\infty$. We mainly consider the question whether $\mathcal{X}_S (k_\infty)$ is a non-abelian free pro-$p$ group or not. In the former part, we treat the case when $k$ is an imaginary quadratic f…
Let $K$ be an imaginary quadratic field, $N$ be a positive integer, $f(z)$ be a newform of level $Γ_1(N)$, and $A_f$ be the abelian variety associated to $f$. For each $τ\in K$ ($\operatorname{Im} τ>0$), we construct a certain point $P_τ$ on $A_f$ defined over an extended ring class field of $K$ of level $N$. Our construction generalizes Birch's construction of the Heegner points to the abelian varieties associated to modular forms of level $Γ_1(N)$ and nontrivial character. Then, we show that $P_τ$'s satisfy the distribution and congruence relations of an Euler system, which implies that it s…
Let $\ell$ and $p$ be odd primes. For a positive integer $μ$ let $k_μ$ be the ray class field of $k=\mathbb{Q}(e^{2πi/\ell})$ modulo $2p^μ$. We present certain class fields $K_μ$ of $k$ such that $k_μ\leq K_μ\leq k_{μ+1}$, and find the degree of $K_μ/k_μ$ explicitly. And we also construct, in the sense of Hilbert, primitive generators of the field $K_μ$ over $k_μ$ by using Shimura's reciprocity law and special values of theta constants.
Generation of Hecke fields by squares of cyclotomic twists of modular $L$-values
本文证明非CM椭圆新形式及其分圆扭曲的Hecke域复合可由中心L值绝对值平方生成，方法基于单幂混合猜想解决技术。
External Difference Families Arising from Two or Three Cyclotomic Classes
本文研究有限域中由分圆类构成的外部差族，证明素数$p=12k+1$时三块族$\{C_0^6,C_2^6,C_4^6\}$构成EDF当且仅当$k$为平方数，并恢复了两块情形的经典刻画。
Let $K = \mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$, and let $\mathcal{O}$ be the ring of integers of $K$. The prime $2$ splits in $K$, say $2\mathcal{O} = \mathfrak{p} \mathfrak{p}^\ast$, and there is a unique $\mathbb{Z}_2$-extension $K_\infty$ of $K$, which is unramified outside $\mathfrak{p}$. Let $H$ be the Hilbert class field of $K$, and write $H_\infty = HK_\infty$. Let $M(H_\infty)$ be the maximal abelian $2$-extension of $H_\infty$, which is unramified outside the primes above $\mathfrak{p}$, and put $X(H_\infty) = \mathrm{Gal}(M(H_\infty)/H_\inf…
We investigate the complex spectra \[ X^{\mathcal A}(β)=\left\{\sum_{j=0}^na_jβ^j : n\in{\mathbb N},\ a_j\in{\mathcal A}\right\} \] where $β$ is a quadratic or cubic Pisot-cyclotomic number and the alphabet $\mathcal A$ is given by $0$ along with a finite collection of roots of unity. Such spectra are discrete aperiodic structures with crystallographically forbidden symmetries. We discuss in general terms under which conditions they possess the Delone property required for point sets modeling quasicrystals. We study the corresponding Voronoi tilings and we relate these structures to quasilatti…
Let p be an odd prime. Let K = \Q(zeta) be the p-cyclotomic number field. Let v be a primitive root mod p and sigma : zeta --> zeta^v be a \Q-isomorphism of the extension K/\Q generating the Galois group G of K/\Q. For n in Z, the notation v^n is understood by v^n mod p with 1 \leq v^n \leq p-1. Let P(X) = \sum_{i=0}^{p-2} v^{-i}X^i \in \Z[X] be the Stickelberger polynomial. P(sigma) annihilates the class group C of K. There exists a polynomial Q(X) \in \Z[X] such that P(sigma)(sigma-v) = p\times Q(sigma) and such that Q(sigma) annihilates the p-class group C_p of K (the subgroup of exponent p…
On $\mathbb{Z}_2$-extensions of real quadratic fields
本文通过迭代扩充分圆单位群，推导了实二次域 $\mathbb{Z}_2$-扩张各层类数中2的幂指数序列的若干规律。
Let $p$ be an odd prime number and $k$ an imaginary quadratic field in which $p$ splits. In this paper, we consider a weak form of Greenberg's generalized conjecture for $p$ and $k$, which states that the non-trivial Iwasawa module of the maximal multiple $\mathbb{Z}_p$-extension field over $k$ has a non-trivial pseudo-null submodule. We prove this conjecture for $p$ and $k$ under the assumption that the Iwasawa $λ$-invariant for a certain $\mathbb{Z}_p$-extension over a finite abelian extension of $k$ vanishes and that the characteristic ideal of the Iwasawa module associated to the cyclotomi…
On $\mathbb{Z}_2$-extensions of real quadratic fields with class group of $2$-rank three
本文给出非分歧Iwasawa模有限性判据，构造无穷族实二次域，其类群$2$-秩为三且分圆$\mathbb{Z}_2$-扩张完全分歧并满足Greenberg猜想。
We prove a slight generalization of Iwasawa's `Riemann-Hurwitz' formula for number fields and use it to generalize Ferrero's and Kida's well-known computations of Iwasawa λ-invariants for the cyclotomic Z_2-extensions of imaginary quadratic number fields. In particular, we show that if p is a Fermat prime, then similar computations of Iwasawa λ-invariants hold for certain imaginary quadratic extensions of the unique subfield k of Q(ζ_{p^2}) such that [k:Q] = p. In fact, we actually prove more by explicitly computing cohomology groups of principal ideals. The computation of lambda invariants ob…
On the Existence of the Maximal Unramified Pro-$2$-Extension over the Cyclotomic $\mathbb{Z}_2$-Extension with Prescribed Metacyclic Galois Group
研究特定metacyclic非模群作为数域最大无分歧2-扩展的Galois群的可能性，并提出新方法探讨Greenberg猜想。
On $k$-free numbers in cyclotomic fields: entropy, symmetries and topological invariants
本研究将二次域上的$k$-无幂数相关动力系统研究拓展至分圆域，探讨其熵与扩展对称性，揭示动力学与数论概念的相互作用，丰富了数论点集的拓扑动力系统研究。
Iwasawa invariants and class number parity of multi-quadratic number fields
本文基于Iwasawa和Kida的方法，通过详细研究Hasse单位指标和素理想分歧，得到数域分圆Z_2扩张的Iwasawa不变量λ_2的显式结果，并在Greenberg猜想下给出虚多二次域λ_2的显式公式，进而提出判定多二次域类数奇偶性的准则。
On automorphisms of some semidirect product groups and ranks of Iwasawa modules
本文受 Kundu-Washington 启发，研究了类反分圆 ${\Bbb Z}_p$-扩张，证明了特定该类扩张的 $λ$ 值恒为偶数，且 CM 域相应扩张理想类群的 $p$-部分恒为非循环。
We construct étale generalized Heisenberg group covers of hyperelliptic curves over number fields. We use these to produce infinite families of quadratic extensions of cyclotomic fields that admit everywhere unramified generalized Heisenberg Galois extensions.
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。