# Questions tagged [iwasawa-theory]

The iwasawa-theory tag has no usage guidance.

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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}} ^\...

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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 ...

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233
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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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221
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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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154
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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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### Tate module of elliptic curves; Commuting Hom functor and tensor product in the second coordinate

Let $\Lambda$ be the Iwasawa Algebra of the Galois group of the cyclotomic $\mathbb{Z}_p$-extension $\mathbb{Q}_{cyc}$ of $\mathbb{Q}$. Let $\widehat{\Lambda}$ be its Pontryagin dual (i.e the ...

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### Elementary Iwasawa module

Let $k$ be a given number field. What is the importance and applications of knowing that the Iwasawa module $X_\infty$ of $k$ is an elementary $\Lambda$-module?