William G. McCallum
In this paper we prove Greenberg's pseudo-null conjecture for the field of p-th roots of unity in the case that p exactly divides the class number and the index of the global units in the local units. We also generalize to the case of multiple Z_p-extensions a theorem of Iwasawa on the Kummer extension of the cyclotomic Z_p-extension generated by p-power roots of units .
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We describe Greenberg's pseudo-null conjecture, and prove a result describing conditions under which the pseudo-null conjecture for a number field $K$ implies the conjecture for finite extensions of $K$. We then apply the result to the cyclotomic $\mathbb{Z}_p$-tower above a cyclotomic field of prime roots of unity, verifying the conjecture for a large class of cyclotomic fields.
Let $K/ \mathbb{Q}$ be an imaginary $S_3$-extension, and $p$ a prime number which splits into exactly three primes in $K$. We give a sufficient condition for the validity of Greenberg's generalized conjecture for $K$ and $p$.
We investigate the group of universal norms attached to the cyclotomic Z {\ell}-tower of a totally real number field in connection with Grenberg's conjecture on Iwasawa invariants of such a field.
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。
In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.
For an algebraic number field $K$ and a prime number $p$, let $\widetilde{K}/K$ be the maximal multiple $\mathbb{Z}_p$-extension. Greenberg's generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-$p$ extension of $\widetilde{K}$ is pseudo-null over the completed group ring $\mathbb{Z}_p[\![\mathop{\mathrm{Gal}}\nolimits(\widetilde{K}/K)]\!]$. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers.
We use logarithmic {\ell}-class groups to take a new view on Greenberg's conjecture about Iwasawa {\ell}-invariants of a totally real number field K. By the way we recall and complete some classical results. Under Leopoldt's conjecture, we prove that Greenberg's conjecture holds if and only if the logarithmic classes of K principalize in the cyclotomic Z{\ell}-extensions of K. As an illustration of our approach, in the special case where the prime {\ell} splits completely in K, we prove that the sufficient condition introduced by Gras just asserts the triviality of the logarithmic class group …
For a prime number $p$ and a number field $k$, let $\tilde{k}$ be the compositum of all $\mathbb{Z}_p$-extensions of $k$. Greenberg's Generalized Conjecture (GGC) claims the pseudo-nullity of the unramified Iwasawa module $X(\tilde{k})$ of $\tilde{k}$. It is known that, when $k$ is an imaginary quadratic field, GGC has a consequence on the Iwasawa invariants associated to $\mathbb{Z}_p$-extensions of $k$. In this paper, we partially generalize it to arbitrary number fields $k$.
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 $p$ 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…
We formulate a general conjecture on the characteristic polynomials of S-decomposed T-ramified Iwasawa modules over the cyclotomic Z {\ell}-extension of a number field. We show that this conjecture is equivalent to the conjunctions of the classical conjectures of Leopoldt and of Gross-Kuz'min. We so extend a result of semi-simplicity of Greenberg and, by the way, an isomorphism of Kuz'min.
In this paper we prove one side divisibility of the Iwasawa-Greenberg main conjecture for Rankin-Selberg product of a weight two cusp form and an ordinary CM form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $\mathrm{GU}(3,1)$, generalizing earlier result of the third-named author to allow non-ordinary cusp forms. The main result is a key input in the third author's proof for Kobayashi's $\pm$-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semi-ordinary Hida theory along an appropriate smaller weight space, …
Let $k$ be a totally real number field and $p$ a prime. We show that the ``complexity'' of Greenberg's conjecture ($λ= μ= 0$) is of $p$-adic nature governed (under Leopoldt's conjecture) by the finite torsion group ${\mathcal T}_k$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $k$, by means of images in ${\mathcal T}_k$ of ideal norms from the layers $k_n$ of the cyclotomic tower (Theorem (5.2)). These images are obtained via the formal algorithm computing, by ``unscrewing'', the $p$-class group of~$k_n$. Conjecture (5.4) of equidistribution of these images would…
On Greenberg's conjecture for rational elliptic curves at Eisenstein primes
本文证明了有理椭圆曲线在Eisenstein素数处关于Selmer群代数Iwasawa μ-不变量的Greenberg猜想，并通过Iwasawa主猜想确认了Stevens关于解析μ-不变量的猜想。
Greenberg's conjecture and Iwasawa module of Real biquadratic fields II
本文研究实双二次域的$2$-类群秩稳定性，验证了部分数域的格林伯格猜想，并确定了具有平凡$2$-Iwasawa模的实双二次域列表。
Pre-print of a publication in "Annales mathématiques du Qu{é}bec". Let $k$ be a totally real number field and let $k_\infty$ be its cyclotomic $\mathbb{Z}_p$-extension for $p$ totally split in $k$. This text completes our article entitled: "Approche $p$-adique de la conjecture de Greenberg pour les corps totalement réels" (Annales Mathématiques Blaise Pascal 2017), by means of heuristics on the $p$-adic behavior of the norms, in $k_n/k$, of the ideals in $k_\infty$ ; indeed, this conjecture (on the nullity of the invariants $λ$ et $μ$ of Iwasawa) depends of images in the torsion group ${\mathc…
On Greenberg's generalized conjecture for families of number fields
本文研究 Greenberg 广义猜想，证明对虚数域而言，特殊 $\mathbb{Z}_p^2$-扩张上的若干伪零性条件可推出该猜想，并指出这些条件被部分数域族满足。
Greenberg's conjecture and Iwasawa module of Real biquadratic fields I
本文研究实双二次域的Greenberg猜想，提出并探讨使Iwasawa模的秩与第一层类群的秩相等的问题，并构造了多族使Iwasawa模平凡或同构于特定循环群及直积的实双二次域，同时给出实三二次域类群2-秩的相关结果。
For a number field $F$ and an odd prime number $p,$ let $\tilde{F}$ be the compositum of all $\mathbb{Z}_p$-extensions of $F$ and $\tildeΛ$ the associated Iwasawa algebra. Let $G_{S}(\tilde{F})$ be the Galois group over $\tilde{F}$ of the maximal extension which is unramified outside $p$-adic and infinite places. In this paper we study the $\tildeΛ$-module $\mathfrak{X}_{S}^{(-i)}(\tilde{F}):=H_1(G_S(\tilde{F}), \mathbb{Z}_p(-i))$ and its relationship with $X(\tilde{F}(μ_p))(i-1)^Δ,$ the $Δ:=\mathrm{Gal}(\tilde{F}(μ_{p})/\tilde{F})$-invariant of the Galois group over $\tilde{F}(μ_{p})$ of the …
Let k be a totally real number field ant let k$\infty$ be its cyclotomic Zp-extension for a prime p\textgreater{}2. We give (Theorem 3.2) a sufficient condition of nullity of the Iwasawa invariants lambda, mu, when p totally splits in k, and we obtain important tables of quadratic fields and p for which we can conclude that lambda = mu=0.We show that the number of ambiguous p-classes of kn (nth stage in k$\infty$) is equal to the order of the torsion group T, of the Galois group of the maximal Abelian p-ramified pro-p-extension of k (Theorem 4.2), for all n \textgreater{}\textgreater{} e, wher…
An Analogue of Greenberg's Conjecture for CM Fields
本文在Leopoldt猜想成立的假设下，证明CM域存在唯一的S-分歧Zp-扩张，推导了其Iwasawa不变量μ=λ=0的类似Greenberg准则，并给出虚双二次域的对应数值判别准则。
Greenberg's $μ=0$ conjecture for lisse sheaves over global function fields
本文在正特征下证明了Greenberg的μ=0猜想的类比，针对全局函数域上的lisheaves，展示了Selmer群的结构及相关变形环的性质。
In this paper we prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points when the global sign is -1, using a recent work of the author on one divisibility of Iwasawa-Greenberg main conjecture for Rankin-Selberg p-adic L-functions. As a byproduct we also prove the equality for the above mentioned main conjecture under some local conditions.
We show that the cyclotomic Iwasawa--Greenberg Main Conjecture holds for a large class of modular forms with multiplicative reduction at $p$, extending previous results for the good ordinary case. In fact, the multiplicative case is deduced from the good case through the use of Hida families and a simple Fitting ideal argument.
Remarks on Greenberg's conjecture for Galois representations associated to elliptic curves
本文研究椭圆曲线 Galois 表示的 Greenberg 猜想，借助 Coates–Sujatha 精细 Selmer 群定理，在若干 Galois 理论条件下证明可约情形结论，并讨论不可约情形的 μ 不变量消失。
A Study of Fine Selmer Groups Over Function Fields via Greenberg Neighbourhoods
本文利用Kleine的技巧，研究了特征为$p$的全局函数域$K$上椭圆曲线精细Selmer群的Iwasawa不变量在不同$\mathbb{Z}_p$-扩张间的关系，并将其与Coates和Sujatha的Conjecture A的类比相联系。
We prove a "discrete analogue" for Taelman's class modules of certain Conjectures formulated by R. Greenberg for cyclotomic fields.
We prove under mild hypotheses the three-variable Iwasawa main conjecture for $p$-ordinary modular forms in the indefinite setting. Our result is in a setting complementary to that in the work of Skinner-Urban, and it has applications to Greenberg's nonvanishing conjecture for the first derivatives at the center of $p$-adic $L$-functions of cusp forms in Hida families with root number $-1$ and to Howard's horizontal nonvanishing conjecture.
We prove the exceptional zero conjecture for the symmetric powers of CM cuspidal eigenforms at ordinary primes. In other words, we determine the trivial zeroes of the associated p-adic L-functions, compute the L-invariants, and show that they agree with Greenberg's L-invariants.
Mazur's growth number conjecture and congruences
本文研究虚二次域上 Mazur 关于 Selmer 秩增长的猜想在 Galois 表示 $p$-同余下的保持性，证明了对特定三元组及模形式 Greenberg Selmer 群的类似结果，并得出非反圆扩张中 Mordell-Weil 秩的有界性。
On the Iwasawa invariants of Kato's zeta elements for modular forms
研究Kato主猜想中不依赖p-进L函数的Iwasawa模的不变量行为，推广Greenberg-Vatsal等人的工作，并在Katoζ元素模p非零的假设下建立高权重模形式的主猜想传播。
In this paper we propose and prove an anticyclotomic Iwasawa main conjecture for Heegner points on supersingular elliptic curves with $a_p=0$. The result has a "$\pm$" nature in the sense of Kobayashi. The proof uses a recent work of the second author on one divisibility in the Iwasawa--Greenberg main conjecture for Rankin-Selberg $p$-adic $L$-functions, together with an argument of Howard (adapted to our "$\pm$"-situation). As a byproduct, we also obtain an improvement of Skinner's result on a converse to the Gross--Zagier--Kolyvagin theorem.
We prove, assuming Greenberg's conjecture, that the ordinary eigencurve is Gorenstein at an intersection point between the Eisenstein family and the cuspidal locus. As a corollary, we obtain new results on Sharifi's conjecture. This result is achieved by constructing a universal ordinary pseudodeformation ring and proving an $R = \mathbb T$ result.
Let $A' = \varprojlim_n$ be the projective limit of the $p$-parts of the ideal class groups of the $p$ integers in the $\Z_p$-cyclotomic extension $\K_{\infty}/\K$ of a CM number field $\K$. We prove in this paper that the $T$ part $(A')^-(T) = 0$. This fact has been explicitly conjecture by Kuz'min in 1972 and was proved by Greenberg in 1973, for abelian extensions $\K/\Q$. Federer and Gross had shown in 1981 that $(A')^-(T) = 0$ is equivalent to the non-vanishing of the $p$-adic regulator of the $p$-units of $\K$.
For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the field. We determine the kernel and the cokernel of this map assuming Greenberg's conjecture on the vanishing of the lambda-invariant of the extension.
An equivariant $p$-adic Artin conjecture
本文提出 Greenberg $p$-进 Artin 猜想的等变形式，并借助 Iwasawa 代数全商环的 Wedderburn 分解证明其在多类一维容许 $p$-进 Lie 扩张情形成立。
In this paper we prove that the $p$-adic $L$-function that interpolates the Rankin-Selberg product of a general weight two modular form which is unramified and non-ordinary at $p$, and an ordinary CM form of higher weight contains the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-function. This generalizes an earlier work of the author to the non-ordinary case. The result of this paper plays a crucial role in the proof of Iwasawa main conjecture and refined Birch-Swinnerton-Dyer formula for supersi…
Stability of 2-class groups in the $\mathbb{Z}_2$-extension of certain real biquadratic fields
本文通过初等群论与类域论方法，验证了某无限族实双二次域分圆 $\mathbb{Z}_2$-扩张中各层 $2$-类群的 Greenberg 猜想，并关联了理想类 capitulation 与类群相对大小。
On the unramified Iwasawa module of a $\mathbb{Z}_p$-extension generated by division points of a CM elliptic curve
本文研究由复乘椭圆曲线除点生成的 $\mathbb{Z}_p$-扩张的非分歧 Iwasawa 模，指出 Greenberg 猜想对该扩张失效，并主要探讨其弱形式猜想的类比。
On the Iwasawa Main Conjecture for generalized Heegner classes in a quaternionic setting
本文针对更高权普通模形式与满足松弛Heegner假设的虚二次域，沿用Howard等的方法构造广义Heegner类的$Λ$-进Kolyvagin系统，证明了反循环岩泽主猜想的一个整除关系，并将其应用于Magrone定义的$p$-进$L$-函数的岩泽-格林伯格主猜想。
The Iwasawa $μ$-invariant of certain elliptic curves of analytic rank zero
本文利用 $p$-进 $L$-函数与 Rankin-Selberg 方法的新技术，对若干解析秩为零的椭圆曲线证明了 Greenberg 关于分圆 $\mathbb{Z}_p$-扩张上 Selmer 群 $μ$-不变量为零的猜想的新情形。
The Iwasawa $μ$-invariants of Elliptic Curves over $\mathbb{Q}$
针对 Greenberg 关于椭圆曲线 Selmer 群 $μ$-不变量为零的猜想，本文证明 $\mathbb{Q}$ 上椭圆曲线在除有限多个良好通常约化素数外均满足 $μ\leq 1$。
For primes $q \equiv 7 \mod 16$, the present manuscript shows that elementary methods enable one to prove surprisingly strong results about the Iwasawa theory of the Gross family of elliptic curves with complex multiplication by the ring of integers of the field $K = \mathbb{Q}(\sqrt{-q})$, which are in perfect accord with the predictions of the conjecture of Birch and Swinnerton-Dyer. We also prove some interesting phenomena related to a classical conjecture of Greenberg, and give a new proof of an old theorem of Hasse.
We prove a strong form of the trivial zero conjecture at the central point for the $p$-adic $L$-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of $\mathrm{GL}_2$ over a totally real field, which is Iwahori spherical at places above $p$. In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the $p$-adic $L$-function of a nearly finite slope family and on improved $p$-adic $L$-functions that we construct using automorphic symbols and overconvergent cohomology. For higher order zeros we d…
On the Fontaine-Mazur conjecture for $p=3$
本文在 p=3 情形下证明了二维正则 Galois 表示的 Fontaine-Mazur 猜想剩余情形，从而完成所有奇素数正则情形的证明，主要基于 p-进 Langlands 对应、Galois 形变理论及潜在 pro-模性结果的最新进展。
Let $p$ be an odd prime. We prove the cyclotomic Iwasawa Main Conjecture of K.Kato for the motive attached to an eigencuspform $f\in S_{k}(Γ_{0}(N))$ with arbitrary reduction type at $p$ under mild assumptions on the residual Galois representation $\barρ_{f}$. Under the same hypotheses, we also prove the generalized Iwasawa Main Conjecture for $p$-adic families of modular forms. The Iwasawa Main Conjecture for $f$ is deduced by a limit argument involving fundamental lines from a universal Iwasawa Main Conjecture over the universal deformation space of $\barρ_{f}$, which itself follows from the…
On Conjectures of $μ$-Invariant for Selmer Groups at $p=2$
本文证明了 $p=2$ 时具有次数 2 有理同源的椭圆曲线的 Greenberg 猜想，给出 $2$-adic $\mu$-不变量的精确上界，并证明精细 Selmer 群 $\mu$-不变量消失。
We formulate a multi-variable p-adic Birch and Swinnerton-Dyer conjecture for p-ordinary elliptic curves A over number fields K. It generalises the one-variable conjecture of Mazur-Tate-Teitelbaum, who studied the case K=Q and the phenomenon of exceptional zeros. We discuss old and new theoretical evidence towards our conjecture and in particular we fully prove it, under mild conditions, in the following situation: K is imaginary quadratic, A=E_K is the base-change to K of an elliptic curve over the rationals, and the rank of A is either 0 or 1. The proof is naturally divided into a few cases.…
A famous conjecture bearing the name of Vandiver states that $p \nmid h_p^+$ in the $p$ - cyclotomic extension of $\Q$. Heuristics arguments of Washington, which have been briefly exposed in Lang (1978), p. 261 and Washington (1996), p. 158 suggest that the Vandiver conjecture should be false if certain conditions of statistical independence are fulfilled. In this note, we assume that Greenberg's conjecture is true for the \nth{p} cyclotomic extensions and prove an elementary consequence of the assumption that Vandiver's conjecture fails for a certain value of $p$: the result indicates that th…
Iwasawa Main Conjecture for Supersingular Elliptic Curves and BSD conjecture
本文将 Kobayashi 的 ±-岩泽主猜想归约并证明为岩泽–Greenberg 主猜想，结合广义局部理论与 Beilinson–Flach 显式互反律，推出超奇异椭圆曲线结果及 BSD 公式 p-部分，并给出可证明完整 BSD 的无 CM 椭圆曲线无限族。
A Heuristic approach to the Iwasawa theory of elliptic curves
本文从统计角度研究Greenberg的μ=0猜想，扩展Poonen和Rains的启发法以提供进一步证据，通过Iwasawa模的交集检测μ不变量的消失。
Leopoldt's Conjecture is a statement about the relationship between the global and local units of a number field. Approximately the conjecture states that the Z_p-rank of the diagonal embedding of the global units into the product of all local units equals the Z-rank of the global units. The variation we consider asks: Can we say anything about the Z_p-rank of the diagonal embedding of the global units into the product of some local units? We use the p-adic Schanuel Conjecture to answer the question in the affirmative and moreover we give a value for the Z_p-rank (of the diagonal embedding of …
Recent attempts at studying the Fermat equation over number fields have uncovered an unexpected and powerful connection with $S$-unit equations. In this expository paper we explain this connection and its implications for the asymptotic Fermat conjecture.
Colmez conjectured a formula relating the Faltings height of CM abelian varieties to a certain linear combination of log derivatives of $L$-functions. In this paper, we study the case of unitary CM fields and by studying the class functions that arise, we reduce the conjecture to a special case. Using the Galois action, we prove more cases of the Colmez Conjecture.
For an odd prime number $p$, we study the number of generators of the unramified Iwasawa modules of the maximal multiple $\mathbb{Z}_p$-extensions over Iwasawa algebra. In a previous paper of the authors, under several assumptions for an imaginary quadratic field, we obtain a necessary and sufficient condition for the Iwasawa module to be cyclic as a module over the Iwasawa algebla. Our main result is to give methods for computation and numerical examples about the results. We remark that our results do not need the assumption that Greenberg's generalized conjecture holds.
Local-global compatibility and the exceptional zero conjecture for GL(3)
本文研究 GL(3) 中 p 处 Steinberg 的 p-常自守表示，通过算术上同调证明自守例外零点猜想，并借助 GL_n 挠类在 p 处的局部-整体相容性推出 L-不变量相等。
We prove an old conjecture of Erd{\H o}s and Graham on sums of unit fractions: There exists a constant $b>0$ such that if we $r$-color the integers in $2,b^r]$, then there exists a monochromatic set $S$ such that $\sum_{n \in S} 1/n=1$.
Let p be an odd prime satisfying Vandiver's conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z_p-extensions of the pth cyclotomic field and the Galois group G of the unramified pro-p extension of the cyclotomic field of all p-power roots of unity. We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for G to be abelian. The bound and the condition in the two results are given in terms of…
Rubin's conjecture on local units in the anticyclotomic tower at inert primes: $p=3$ case
本文通过推广Burungale-Kobayashi-Ota的工作，证明了$p=3$情形下局部单位在反分圆塔中结构的Rubin猜想。
In this article, we introduce a systematic new method to investigate the conjectural p-adic meromorphic continuation of Professor Bernard Dwork's unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p.
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