Questions tagged [iwasawa-theory]
The iwasawa-theory tag has no usage guidance.
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Is there something I am missing about the computation of the $p$-part of the class groups of cyclotomic fields?
Well, the answer of the question in the title in certainly Yes, many things in fact, but let me be more precise.
In 1958, Serre gave a Bourbaki talk on the recent works of Iwasawa on class groups in ...
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How is Taylor-Wiles patching "horizontal Iwasawa theory"?
I have recently been reading into the proof of modularity of semistable elliptic curves, in particular (what is now known as) the Taylor-Wiles patching argument used to prove the $R=T$ theorem in the ...
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On Iwasawa theory of elliptic curves in $\mathrm{PGL}_2(\mathbb{Z}_p)$-extension
Let $E$ be an elliptic curve over the rationals $\mathbb{Q}$. We consider the Galois representation attached to $E$ by acting on its $p$-adic Tate module $T_p(E)$,
$$
\rho_E: G_{K} \rightarrow \mathrm{...
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Maximal p-extension and pro-p extension
I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help.
Q_1: About terminology $p$-extension.
I find many reference use maximal $p$-extension or maximal abelian p-extension ...
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Iwasawa theory and an $M$ finitely generated $\Lambda _{\mathcal{O}}$ module
A bit of context: I am currently self studying Iwasawa theory, and faced the following theorem:
Let $M$ be a finitely generated $\Lambda _{\mathcal{O}}$ module, then
$$M \sim \Lambda _{\mathcal{O}} ^\...
user516405
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Clarifications about the Iwasawa Main Conjecture
I would like to clarify a couple of things regarding the Iwasawa main conjecture.
In the paper where Mazur and Wiles prove the main conjecture, on page 182, it is written that $h_p(\omega^i, T)$ is ...
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Integration against Eisenstein series can be regarded as a cup product
This summer, I was very fortunate and honored to attend the conference "Iwasawa 2023" at the University of Cambridge as a young Ph.D. student on Iwasawa theory. There, one of the speakers, ...
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Why does passing to a $\mathbf{Z}_p$-extension make things easier?
In Iwasawa theory, even if one is only interested in questions about a number field $K$ (e.g. class groups of $\mathbf{Q}(\mu_p)$, Selmer groups of abelian varieties over $\mathbf{Q}$), to prove deep ...
user471019
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On Kato's proof that Iwasawa Cohomology is free
In Kato's $p$-adic Hodge Theory and Values of Zeta Functions of Modular Forms, he proves in 13.8 that the first Iwasawa cohomology:
$$\mathbf{H}^1(T):=\varprojlim_nH^1(\mathbb{Z}[ζpn,1/p],T)$$
is free ...
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Compositum of field extensions in context of $\mathbb Z_p$ extension
I had asked this question on stackexchange and I think it is better suited for this site.
Suppose I have a $\Gamma \simeq \mathbb Z_p $ extension $F_\infty /F$ of a number field $F$. Let $F_n$ be the ...
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Examples of $\mathbb{Z}_p$-extensions and two $\mathbb{Z}_p$-extensions with a "nontrivial" intersection
Let $k$ be a number field (a finite extension of $\mathbb{Q}$). Let $p$ be a prime. By saying "$\mathbb{Z}_p$-extensions", we mean Galois extensions $K/k$ of Galois group isomorphic to $\...
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Structure theorem for Iwasawa modules over $p$-adic rings of integers
Let $K/\mathbb Q_p$ be a finite extension, and $\mathcal O_K$ the ring of integers of $K$. I am asking for a reference for a structure theorem
of finitely generated modules over the completed group ...
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Quantitative lifting for mod-p elliptic curves to characteristic zero CM elliptic curves
Let $A$ be a supersingular elliptic curve over $\mathbb{Z}/p\mathbb{Z}$ and $\mathcal{O}$ an order in an imaginary quadratic field contained in the quaternion algebra $\operatorname{End}(A)$, then by ...
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Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why?
I am a new learner of Iwasawa theory and currently reading the famous paper by Skinner-Urban in 2014, and the following-up works by many other people. When reading these papers, I found that some ...
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$p$-adic $L$-functions and congruence of $L$-values
I am reading about $p$-adic $L$-functions and I have one question in mind.
To start with, I will write a proof I've learned of a congruence of $L$-values:
Theorem: Let $p\geq5$ be a prime, $\alpha\...
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Invariants of Iwasawa modules
Let $\Lambda$ denote Iwasawa algebra $\mathbb{Z}_p[[\Gamma]]$, where $\Gamma$ is a group isomorphic to $\mathbb{Z}_p$(ring of $p$-adic integers). The structure theorem of the Iwasawa module says: If M ...
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Why $p$-adic measures?
I'm currently learning about the Kubota–Leopoldt $p$-adic $L$-function and I'm noticing that many people view the Kubota–Leopoldt $p$-adic $L$-function as a measure as opposed to a $p$-adic analytic ...
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Generalisation of Sharifi's conjecture for Siegel varieties
I recently learned Sharifi's conjecture in this article by F.Takako and K.Kato.
According to my understanding, this conjecture states that (the conjugation invariant) of the projective limit $\...
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A question about generalized harmonic numbers modulo $p$
Let $p \equiv 1 \pmod{3}$ be a prime and denote $H_{n,m} = \sum_{k = 1}^n 1/k^m$ as the $n,m$-th generalized harmonic number. I'm interested in computing $H_{(p-1)/3,\, 2}$ and $H_{(p-1)/6,\,2}$ ...
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$\mathbb{Z}_\ell$-extensions of global function fields
Given a number field $F$ and a prime $p$, it is natural to study Iwasawa theory over the cyclotomic $\mathbb{Z}_p$-extension of $F$, i.e., the unique $\mathbb{Z}_p$-extension of $F$ which is contained ...
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Extending the analogy between cyclotomic units and elliptic units
There is a nice analogy between cyclotomic units and elliptic units given as follows:
Cyclotomic units are related to special values of the Riemann Zeta function. This is because the logarithmic ...
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question about Sinnott's proof of the Ferrero-Washington Theorem
I'm currently reading the paper "On the $\mu$-invariant of the Γ-transform of a rational function" by W Sinnott. In this paper, he gives an alternate proof that $\mu=0$ for abelian number ...
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A confusion about power series and p-adic measures
In page 16 of these notes on $p$-adic $L$-functions, it makes the following claim:
Let $\alpha$ be a $p$-adic measure on $\mathbf{Z}_p$ which corresponds to a power series $F_{\alpha}(T) \in \mathbf{...
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Describing the Gamma-transform explicitly in terms of power series
The Gamma transform of a measure is defined as follows. If $\alpha$ is a $\mathbf{Z}_p$-valued measure on $\mathbf{Z}_p$, then the Gamma transform of $\alpha$ is:
$$\Gamma_{\alpha}(s) = \int_{\mathbf{...
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Alternative formulation of the Ferrero-Washington Theorem
The Ferrero-Washington theorem says that if $K/\mathbf{Q}$ is an abelian extension, then the cyclotomic $\mathbf{Z}_p$ extension $K^{\text{cyc}}/K$ has $\mu=0$.
In the paper "Iwasawa invariants ...
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What's the class group of $\mathbb{Q}^{\mathrm{ab}}$?
If $K/\mathbb{Q}$ is an infinite algebraic extension, define as usual the class group $Cl_K$ by the direct limit via the natural (conorm) map $Cl_K := \lim\limits_{\rightarrow} Cl_F$,
where $F$ runs ...
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The Gamma-transform and $p$-adic $L$-functions
I'm currently reading the paper "On the $\mu$-invariant of the $\Gamma$-transform of a rational function" by W Sinnott. In this paper, he gives an alternate proof that $\mu=0$ for abelian ...
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Studying a connection between Iwasawa theory and the $K(n)$-local Picard group by some geometry on the Lubin-Tate stack
Let $Pic_n^0$ denote the even part of the $K(n)$-local Picard group, and let $Pic_n^*$ denote $Hom(Pic_n^0, W(\mathbb{F}_{p^n})^x)$. Denote by $L$ the profinite group ring $\mathbb{Z}_p[[Pic_n^*]] $. ...
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Mordell-Weil rank growth in Iwasawa tower
This is more of a reference request in case anyone can direct me to the right literature. I asked originally on MathStack, but I was suggested to better post it here.
If you have an elliptic curve $E/\...
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Order of $37$-Sylow subgroup of ideal class group of $K_{37} = \Bbb Q(\mu_{37^{n}})$ is known to be $37^n$
I want to examine nontrivial examples of what we call Iwasawa class formula,
$c(n)=\mu p^n + \lambda n + \nu$, where $\lambda, \mu \in \mathbf N$ and $\nu \in \mathbf Z$ are parameters depending only ...
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Finding a certain value of $\Gamma_p$
Let $\Gamma_p : \mathbb{Z}_p \to \mathbb{Z}_p^{\times}$ be the $p$-adic gamma function. I thought that I had successfully calculated $\Gamma_p(1 - 1/4)$, but sage is telling me I'm wrong (this is ...
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Value of the quadratic Gauss sum viewed in $\mathbb{C}_p$
Let $p > 2$ be a prime and $q = p^r$ for some $r \in \mathbb{Z}^+$. I will assume that all roots of unity lie in $\mathbb{C}_p^{\times}$. Let $\zeta$ a primitive $p$-th root of unity. Let $Tr : ...
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Non-commutative knot invariants
$\newcommand{\ab}{\mathrm{ab}}$Let $L=K_1\cup \dots \cup K_r$ be a link embedded in a 3-sphere. Here, $K_1,\dots, K_r$ are the component knots of $L$. A prototypical invariant associated with $L$ is ...
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Question about infinitude of $m$-irregular primes
Let $p>2$ be a prime, $\zeta$ a primitive $p$th root of unity, and $m >0$ a square-free integer such that $m \not\equiv 3 \mod 4$ and $\gcd(m,p)=1$. Let $\chi$ be the imaginary quadratic ...
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Does $\mu=0$ for an imaginary quadratic field $K$ imply $\mu=0$ for $\mathbf{Q}$?
Suppose that $E/\mathbf{Q}$ is an elliptic curve and $K$ is an imaginary quadratic field. Let $\mathbf{Q}_{\infty}$ denote the cyclotomic $\mathbf{Z}_p$ extension of $\mathbf{Q}$, and let $K_{\infty}$ ...
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Relationship between fine Selmer groups and class groups
Given an elliptic curve $E_{/\mathbb{Q}}$ and an odd prime number $p$, let $S$ be the set of primes consisting of $p$ and the primes at which $E$ has bad reduction. The fine Selmer group $R_{p^\infty}(...
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Calculating the Galois cohomology group $H^1(K_v, \, E[p^{\infty}])$
Suppose $K$ is a number field and $E$ is an elliptic curve defined over $K$. My question is: how do you compute the local cohomology group $H^1(K_v, \, E[p^{\infty}])$?
As to why I'm asking this, it ...
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Structure theorem for finitely generated $\Lambda$-modules - uniqueness part
In Iwasawa theory, one of the fundamental results is the following structure theorem for finitely generated modules over the ring $\Lambda = \mathbf{Z}_p[[T]]$.
If $M$ is a finitely generated torsion ...
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Structure of finitely generated $\mathbb{Z}/p^n\mathbb{Z}[[S,T]]$-modules
Let $\Omega=\mathbb{Z}/p\mathbb{Z}[[S,T]]$. $\Omega$ is a commutative, Noetherian and integrally closed domain of Krull dimension 2. According to Bourbaki's commutative algebra VII $\S 4$, if $M$ is a ...
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What do congruences between modular forms tell us about $\mu$-invariants of elliptic curves?
This question is based off these notes by Preston Wake about Iwasawa invariants and Hida Families. In the notes, the author asks "why" the elliptic curve $11A3$ has $\mu$-invariant equal to $...
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Classification of cyclotomic fields with class number 1
1.Let $p$ be an odd prime number. Is there a classification of cyclotomic fields of the form $\mathbb{Q}(\zeta_p)$ with class number 1?
2.Is there such a classification for general cyclotomic fields $...
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Has there been much research on the Iwasawa theory of bi-quadratic fields?
The simplest non-trivial example of $\mathbb{Z}_p$-extensions happen over imaginary quadratic fields and I've seen a good amount of research done here (from my understanding still a lot to learn). I ...
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Why is the regulator of the degree 11 extension of $\mathbb{Q}$ so large?
Let $K_\infty/\mathbb{Q}$ denote the $\hat{\mathbb{Z}}$-extension of $\mathbb{Q}$. Then for each $n\geq1$, $K_\infty$ has a unique subfield $K_n$ of degree $n$ over $\mathbb{Q}$.
The fields $K_n$ are ...
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How hard is it to find the first layer of this basic $\mathbb{Z}_p$-extension?
$\DeclareMathOperator\Gal{Gal}$Let $p$ be a prime number and $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. We know that there is a unique subfield $\mathbb{Q}_1$ of $\mathbb{Q}(\zeta_{p^2})$ ...
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Average size of class groups of cyclotomic fields: three perspectives
Let $K$ be a number field. Let $h(K)$ denote the class number (i.e., the size of the ideal class group) of $K$, $R(K)$ be the regulator of $K$, and $\Delta_K$ the discriminant of $K$.
Let $\mathcal{F}$...
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Existence of non-zero pseudo-null submodules
Let $p$ be a rational prime, and let $\Lambda_d$ be the Iwasawa algebra in $d$ variables, i.e. $\Lambda_d = \mathbb{Z}_p[[T_1, \ldots, T_d]]$. Let $A$ be a finitely generated and torsion $\Lambda_d$-...
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Iwasawa theory over function fields - How do eigenvalues vary in $\mathbb Z_\ell$ towers?
Consider a tower of curves $\dots \to C_n \to C_{n-1} \dots \to C_1$ over $\mathbb F_q$ where $C_n: f(x^{\ell^n},y) = 0$.We can look at the multiset $S_n$ of eigenvalues of the Frobenius $\sigma_q$ on ...
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Finiteness of points over the cyclotomic extension for modular forms
Let $\rho(f):G_\mathbb{Q} \rightarrow GL_2(K_f)$ be the Galois representation attached to some cuspidal modular form $f$ where $K_f$ is a finite extension of $\mathbb{Q}_p$.
Let $V_f$ be the vector ...
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Iwasawa's results about relation between Galois cohomology and principal factorization
Let $K$ be a Galois number field with Galois group and units group $G$ and $U$, respectively. How we can relate the first cohomology group $H^1(G,U)$ to principal factorization in K?
I'd try to find ...
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Merel's theorem on uniform bound for torsion of all elliptic curves
I am reading Silverman's book on Arithmetic of elliptic curves. There he mentions a theorem of Merel (Thm 7.5.1) which reads like this.
Thm: For every integer $d \geq 1$, there is a constant $N(d)$ ...
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