Lorentz groups of cyclotomic extensions

The Shafarevich-Tate group in cyclotomic Z_p-extensions at supersingular primes

We study the asymptotic growth of the p-primary component of the Shafarevich-Tate group in the cyclotomic direction at any odd prime of good supersingular reduction, generalizing work of Kobayashi. This explains formulas obtained by Kurihara, Perrin-Riou, and Nasybullin in terms of Iwasawa invariants of modified Selmer groups.

Modularity of elliptic curves over cyclotomic $\mathbb{Z}_p$-extensions of real quadratic fields

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.

Ideal class groups of cyclotomic number fields III

Using an idea going back to Scholz, we construct unramified abelian extensions of cyclotomic extensions of number fields.

Iwasawa's constant $μ$ vanishes in cyclotomic $\Z_p$-extensions of CM fields

Let $\K$ be a galois CM extension of $\Q$ and $\K_{\infty}$ its cyclotomic $\Z_p$-extension. Let $A_n$ be the $p$-parts of the class groups in the intermediate subfields $\K_n \subset \K_{\infty}$ and $\rg{A} = \varprojlim_n A_n$. We show that the $p$-rank of $\rg{A}$ is finite, which is equivalent to the vanishing of Iwasawa's constant $μ$ for $\rg{A}$. (Currently withdrawn)

分圆扩张中Call-Silverman高度的一个下界

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。

Class numbers in cyclotomic Z_p-extensions

For any prime p and any positive integer n, let B(p,n) denote the n-th layer of the cyclotomic Z_p-extension over the rationals. Based on the heuristics of Cohen and Lenstra, and refined by new results on class numbers of particular B(p,n), we provide evidence for the following conjecture: For all primes p and positive integers n, the ideal class group of B(p,n) is trivial.

关于二次分圆扩张的一个注记

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。

Greenberg's conjecture for real quadratic fields and the cyclotomic $\mathbb{Z}_2$-extensions

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.

Fitting ideals of Class groups in Carlitz-Hayes cyclotomic extensions

We generalize some results of Greither and Popescu to a geometric Galois cover $X\rightarrow Y$ which appears naturally for example in extensions generated by $\mathfrak{p}^n$-torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz-Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in \cite{ABBL}) with Goss $ζ$-function.

Mordell-Weil ranks and Tate-Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions

Let $E$ be an elliptic curve defined over a number field $K$ where $p$ splits completely. Suppose that $E$ has good reduction at all primes above $p$. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a finite extension $F$ of $K$ where $p$ is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell-Weil ranks of $E$ over a subextension of the cyclotomic $\mathbb{Z}_p$-extension are bounded. Furthermore,…

On the asymptotic growth of Bloch-Kato-Shafarevich-Tate groups of modular forms over cyclotomic extensions

We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.

On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions

Let $F$ be a number field unramified at an odd prime $p$ and $F_\infty$ be the $\mathbf{Z}_p$-cyclotomic extension of $F$. Let $A$ be an abelian variety defined over $F$ with good supersingular reduction at all primes of $F$ above $p$. Büyükboduk and the first named author have defined modified Selmer groups associated to $A$ over $F_\infty$. Assuming that the Pontryagin dual of these Selmer groups are torsion $\mathbf{Z}_p[[\mathrm{Gal}(F_\infty/F)]]$-modules, we give an explicit sufficient condition for the rank of the Mordell-Weil group $A(F_n)$ to be bounded as $n$ varies.

Galois groups of some iterated polynomials over cyclotomic extensions

Let $φ_p(z)=(z-1)^p+2-ζ_p$, where $ζ_p\in\bar{\mathbb{Q}}$ is a primitive $p$-th root of unity for some odd prime $p$. Building on previous work, we show that the $n$-th iterate $φ_p^n(z)$ has Galois group $[C_p]^n$, an iterated wreath product of cyclic groups, whenever $p$ is not a Wieferich prime.

反分圆 $\mathbb{Z}_p$-扩张的启发式方法

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。

On tamely ramified Iwasawa modules for the cyclotomic Z_p-extension of abelian fields

Let p be an odd prime, and k_\infty the cyclotomic Z_p-extension of an abelian field k. For a finite set S of rational primes which does not include p, we will consider the maximal S-ramified abelian pro-p extension M_S(k_\infty) over k_\infty. We shall give a formula of the Z_p-rank of Gal(M_S(k_\infty)/k_\infty). In the proof of this formula, we also show that M_{q}(k_\infty)/L(k_\infty) is a finite extension for every real abelian field k and every rational prime q distinct from p, where L(k_\infty) is the maximal unramified abelian pro-p extension over k_\infty.

关于某些分圆 $\mathbb{Z}_2$-扩张的 Hilbert $2$-类域塔

On the Hilbert $2$-class field towers of some cyclotomic $\mathbb{Z}_2$-extensions

本文研究特殊 Dirichlet 域分圆 $\mathbb{Z}_2$-扩张各层的 $2$-类域塔长度、Galois 群结构及 capitulation 问题。

Endomorphisms of abelian varieties, cyclotomic extensions and Lie algebras

We prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero.

$2$-class groups of cyclotomic towers of imaginary biquadratic fields and applications

Let $d$ be a positive square-free integer. In this paper we shall investigate the structure of the $2$-class group of the cyclotomic $\mathbb{Z}_2$-extension of the imaginary biquadratic number field $\mathbb{Q}(\sqrt{d},\sqrt{-1})$. Furthermore, we deduce the structure of the $2$-class group of cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{-d})$.

Signature Ranks of Units in Cyclotomic Extensions of Abelian Number Fields

We prove the rank of the group of signatures of the circular units (hence also the full group of units) of ${\mathbb Q}( ζ_m)^+$ tends to infinity with $m$. We also show the signature rank of the units differs from its maximum possible value by a bounded amount for all the real subfields of the composite of an abelian field with finitely many odd prime-power cyclotomic towers. In particular, for any prime $p$ the signature rank of the units of ${\mathbb Q}( ζ_{p^n})^+$ differs from $φ(p^n)/2$ by an amount that is bounded independent of $n$. Finally, we show conditionally that for general cyclo…

On the structure of the Galois group of the maximal pro-$p$ extension with restricted ramification over the cyclotomic $\mathbb{Z}_p$-extension

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…

能表示为两个立方和的整数

Integers that are sums of two cubes in the cyclotomic $\mathbb{Z}_p$-extension

研究能表示为两个立方和的整数在环论Z_p扩张中的性质及应用。

Rankin-Selberg L-functions in cyclotomic towers, I

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…

某些实二次域的割圆$\mathbb{Z}_{2}$-扩张的岩泽模

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。

关于某些实二次域的分圆$\mathbb{Z}_p$-扩张上椭圆曲线的模性

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。

Ideal class groups of cyclotomic number fields II

We first study some families of maximal real subfields of cyclotomic fields with even class number, and then explore the implications of large plus class numbers of cyclotomic fields. We also discuss capitulation of the minus part and the behaviour of p-class groups in cyclic ramified p-extensions.

通用范数与de Rham表示

Universal norms and de Rham representations

本文证明Iwasawa上同调中通用范数计算从分圆扩张可推广至任意包含它的p-adic扩张，并给出适用于Lubin-Tate扩张的扭曲变体。

Weber's class number problem and $p$-rationality in the cyclotomic $\widehat{\mathbb{Z}}$-extension of $\mathbb{Q}$

Let $K$ be the $N$th layer in the cyclotomic $\widehat{\mathbb{Z}}$-extension $\widehat{\mathbb{Q}}$. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington,\,$\ldots$) analyse the behavior of the $p$-class groups ${\mathcal C}_K$. We revisit this problem, in a more conceptual form, since computations show that the $p$-torsion group ${\mathcal T}_K$ of the Galois groups of the maximal abelian $p$-ramified pro-$p$-extension of $K$ (Tate--Shafarevich group of $K$) is often non-trivial; this raises questions since $\# {\mathcal T}_K = \# {\m…

Growth of torsion of elliptic curves with full 2-torsion over quadratic cyclotomic 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.

分圆扩张中具有好约化的$\mathbb{Q}_p$上椭圆曲线的挠点

Torsion of elliptic curves over $\mathbb{Q}_p$ with good reduction in cyclotomic extensions

本文研究分圆扩张下$\mathbb{Q}_p$上带好约化的椭圆曲线挠点结构问题，对任意素数p及对应分圆扩张情形，完成了相关有理点群挠子群的结构分类。

Nontrivial Galois module structure of cyclotomic fields

We say a tame Galois field extension $L/K$ with Galois group $G$ has trivial Galois module structure if the rings of integers have the property that $\Cal{O}_{L}$ is a free $\Cal{O}_{K}[G]$-module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes $l$ so that for each there is a tame Galois field extension of degree $l$ so that $L/K$ has nontrivial Galois module structure. However, the proof does not directly yield specific primes $l$ for a given algebraic number field $K.$ For…

Arithmetic actions on cyclotomic function fields

We derive the group structure for cyclotomic function fields obtained by applying the Carlitz action for extensions of an initial constant field. The tame and wild structures are isolated to describe the Galois action on differentials. We show that the associated invariant rings are not polynomial.

关于具有指定metacyclic Galois群的 cyclotomic $\mathbb{Z}_2$-扩展的最大无分歧pro-$2$-扩展的存在性

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。

Ray Class Groups of Quadratic and Cyclotomic Fields

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.

Torsion groups of elliptic curves over quadratic cyclotomic fields in elementary abelian 2-extensions

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 $K$.

反分圆Selmer群余挠的平均性质

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。

Singular integers and p-class group of cyclotomic fields

Let $p$ be an irregular prime. Let $K=\Q(ζ)$ be the $p$-cyclotomic field. From Kummer and class field theory, there exist Galois extensions $S/\Q$ of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$. We give an algebraic construction of the subfields $M$ of $S$ with degree $[M:\Q]=p$ and an explicit formula for the prime decomposition and ramification of the prime number $p$ in the extensions $S/K$, $M/\Q$ and $S/M$. In the last section, we examine the consequences of these results for the Vandiver's conjecture. This article is at elementary level on Classic…

Cohomology groups of Fermat curves via ray class fields of cyclotomic fields

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…

某些最大分圆域上的Kummer扩张，Ribet的一个有限性定理及TKND-AVKF域

Some Kummer extensions over maximal cyclotomic fields, a finiteness theorem of Ribet and TKND-AVKF fields

本文推广了Ribet关于阿贝尔簇在最大分圆域上挠点有限性的定理，并给出TKND-AVKF域的新例子。

Ordinary deformations are unobstructed in the cyclotomic limit

The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by Wiles and others. Hida has studied the behaviour of these deformations when one considers the $p$-cyclotomic tower of extensions of the field. In the limit, one obtains a deformation ring $R_\infty$ classifying the ordinary deformations of the (Galois group of) the $p$-cyclotomic extension. We show that if $R_\infty$ is Noetherian and certain adjoint $μ$-inv…

On Weil Numbers in Cyclotomic Fields

In this note, we investigate the p-adic behavior of Weil numbers in the cyclotomic $\mathbb Z\_p$-extension of $\mathbb Q(ζ\_p).$ We determlne the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in Iwasawa Theory.

Analogues of Iwasawa's $μ=0$ conjecture and the weak Leopoldt conjecture for a non-cyclotomic $\mathbb{Z}_2$-extension

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…

Growth of torsion of elliptic curves with odd-order torsion over quadratic cyclotomic 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 $E(K)[2] \approx C_1$ 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 $N=15,20,21,24,27,30,35$.

Galois Groups and Genera of a kind of Quasi-cyclotomic Function Fields

We call a (q-1)-th Kummer extension of a cyclotomic function field a quasi-cyclotomic function field if it is Galois, but non-abelian, over the rational function field with the constant field of q elements. In this paper, we determine the structure of the Galois groups of a kind of quasi-cyclotomic function fields over the base field. We also give the genus formulae of them.

Campos ciclotómicos, tercera versión

This book deals with the elementary theory of cyclotomic fields, cyclotomic function fields, abelian extensions, genus fields, $p$--extensions, Drinfeld modules and class field theory. The text is in Spanish.

Rankin-Selberg L-functions in cyclotomic towers, II

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.

虚二次域的环$\nmathbb{Z}_p$扩张中的$p$-类群

$p$-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。

Primitive Complete Normal Bases for Regular Extensions: Exceptional Cyclotomic Modules

A primitive completely normal element for an extension $\mathbb{F}_{q^n}/\mathbb{F}_{q}$ of Galois fields is a generator of the multiplicative group of $\mathbb{F}_{q^n}$, which simultaneously is normal over every intermediate field of that extension. We are going to prove that such a generator exists when $\mathbb{F}_{q^n}/\mathbb{F}_{q}$ is an 'exceptional' regular extension. In combination with [6] our investigations altogether settle the existence of primitive completely normal bases for any regular extension. An important feature of the class of regular extensions is that they comprise ev…

Radical Extensions for the Carlitz--Hayes Module

Let $L/K$ be a finite extension of congruence function fields. We say that $L/K$ is a {\it radical extension} if $L$ is generated by roots of polynomials $u^{M}-α\in K[u]$, where $u^{M}$ is the action of Carlitz-Hayes. We study a special class of these extensions, the {\it radical cyclotomic} extensions. We prove that any radical cyclotomic extension has order a power of the characteristic of $K$. We also give bounds for the Carlitz-Hayes torsion of these extensions.

具有限制 $p$-分歧的阿贝尔 $p$-扩张与 $\mathbb{Q} (\sqrt{q})$ 的分圆 $\mathbb{Z}_2$-扩张

Abelian $p$-extensions with restricted $p$-ramification and the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q} (\sqrt{q})$

本文利用限制p-分歧的阿贝尔p-扩张研究非分歧Iwasawa模X(K_∞)的结构，给出其有限或有限生成的充分条件，并应用于Q(√q)的分圆Z_2-扩张，得到X(k_∞)为循环模的充要条件及Greenberg猜想成立的充分条件。

On the main conjecture of Iwasawa theory for certain non-cyclotomic $\mathbb{Z}_p$-extensions

Let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$, with ring of integers $\mathcal{O}$ and Hilbert class field $H$. Suppose $p\nmid [H:K]$ is a prime number which splits in $K$, say $p\mathcal{O}=\mathfrak{p}\mathfrak{p}^*$. Let $H_\infty=HK_\infty$ where $K_\infty$ is the unique $\mathbb{Z}_p$-extension of $K$ unramified outside $\mathfrak{p}$. Write $M(H_\infty)$ for the maximal abelian $p$-extension of $H_\infty$ unramified outside the primes above $\mathfrak{p}$, and set $X(H_\infty)=\mathrm{Gal}(M(H_\infty)/H_\infty)$. In this paper, we establish the…

分圆域中有界高度代数整数的计数

Counting Algebraic Integers of Bounded Height in Cyclotomic Fields

本文针对素数幂阶分圆域，给出绝对高度不超过B的代数整数和单位个数的渐近上界。

Iwasawa Main Conjecture for the Carlitz cyclotomic extension and applications

We prove an Iwasawa Main Conjecture for the class group of the $\mathfrak{p}$-cyclotomic extension $\mathcal{F}$ of the function field $\mathbb{F}_q(θ)$ ($\mathfrak{p}$ is a prime of $\mathbb{F}_q[θ]\,$), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a $\mathfrak{p}$-adic $L$-function to prove a close analog of the Ferrero-Washington theorem for $\mathcal{F}$ and to provide informations on the $\mathfrak{p}$-adic valuations of the Bernoulli-Goss numbers $β(j)$ (i.e., on the values of the Goss $ζ$-function at…

$\boldsymbol{\text{有理数域}}$分圆扩张上有理椭圆曲线的挠群

Torsion of Rational Elliptic Curves over the Cyclotomic Extensions of $\mathbb{Q}$

本文研究有理椭圆曲线在有理数域分圆扩张上的挠群分类问题，对任意素数p给出同构分类，p-1不被小整数整除时得到更精细结果，还提供了排除阿贝尔数域上未实现挠结构的工具。

On the structure and stability of ranks of $2$-class groups in cyclotomic $\mathbb{Z}_{2}$-extensions of certain real quadratic fields

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…