Questions tagged [iwasawa-theory]

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properties of anti-cyclotomic extension

Let $K$ be an imaginary quadratic field, and $p$ be a primes number, there exists an unique $\mathbb{Z}_p$-extension of $K$, we denote it by $K_{p,\infty}$ such that the action of complex conjugate $c$...
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173 views

Classification of finitely generated modules over non-commutative rings

Let $\Lambda$ be a commutative integral ring with an automorphism $\sigma$ (I have in mind $\mathbb Z_p[[t]]$ and $\sigma(t) = (1+t)^\alpha - 1$ with $\alpha \in \Lambda^\times$) and $R = \Lambda\{F\}$...
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102 views

Completely split primes in non-anticyclotomic $\mathbb{Z}_p$-extensions

In his colloquium paper "The Structure of Selmer Groups" Greenberg writes the following: If $K$ is an imaginary quadratic field ... it is conjectured that for any [non-anticyclotomic] $\mathbb{Z}_p$...
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Regular growth of ranks in Iwasawa tower

$\newcommand{\rank}{\operatorname{rank}}$Let $G=H \times K$ be a torsion free pro-$p$, $p$-adic Lie group. Let $H =\mathbb{Z}_p$, the ring of $p$-adic integers and $K$ is a non-commutative torsion ...
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156 views

Completed cohomology and variants

I am interested in the following set up: I have an ind-sequence of curves $\dots X_2\to X_1$ defined over a finite field of characteristic $p$ such that $X_n/X_{n-1}$ is a Galois degree $\ell$ cover ...
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269 views

Why can Euler systems constructed from algebraic cycles only be anticyclotomic?

In a footnote to the 2018 Zerbes-Loeffler lecture notes from the Arizona Winter School, it's stated that Euler systems constructed from algebraic cycles "cannot give a full Euler system, only an ...
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144 views

Adjoint Selmer groups and Deformation rings

Let $\rho:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}_2(\bar{\mathbb{Z}}_p)$ be a $p$-adic Galois representation associated to a $p$-ordinary Hecke eigencuspform, let $...
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60 views

Is $\mu=0$ for any $\mathbb{Z}_p$-extension where primes are finitely decomposed?

Let $F/\mathbb{Q}$ be a number field. Iwasawa conjectured that for the cyclotomic $\mathbb{Z}_p$-extension of $F$, the $\mu$-invariant should be 0. Is the same expected to be true for any $\mathbb{Z}...
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156 views

Modular symbols associated to Rankin Selberg convolutions and the symmetric square

I'm interested in understanding how one may associate modular symbols to the L-functions and $p$-adic L-functions associated to the Rankin Selberg convolution of two modular forms/ elliptic curves and ...
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75 views

Generalized Greenberg's Conjecture for Imaginary Quadratic Fields

Let $K$ be an imaginary quadratic field and $\widetilde{K}$ be the compositum of all $\mathbb{Z}_p$ extensions of $K$. Here, $\widetilde{K}/K$ is a $\mathbb{Z}_p^2$ extension. Define $M(\widetilde{K}...
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136 views

Euler characteristics in the rank one case

Suppose $E$ is an elliptic curve over a number field with good ordinary reduction at the primes above a fixed odd prime $p$. We are interested in the Iwasawa theory over the cyclotomic $\mathbb{Z}_p$ ...
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Is there a converse to Vatsal's theorem on congruence of p-adic L-functions?

Let $f=\sum_n a_n(f) q^n$ and $g=\sum_n a_n(g) q^n$ be normalized (cuspidal) newforms whose Fourier coefficients are contained in the p-adic field K for which the uniformizer of $\mathcal{O}_K$ is ...
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Existence of nontrivial finite sub-modules in the cyclotomic extension

It is a well established fact (by Greenberg) that if $p$ is a prime of good ordinary reduction of an elliptic curve $E/\mathbb{Q}$, then the dual of the Selmer group, denoted by $X(E/\mathbb{Q}_{cyc})$...
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Trivial fine Selmer group in the cyclotomic extension

In explicit examples that I have seen worked out, it appears that when the fine Selmer group is finite in the cyclotomic extension it is in fact trivial. Is there any reason to expect that this ...
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480 views

Iwasawa theory and perfectoid spaces

Have there been any applications of perfectoid theory to Iwasawa theory? At a first glance, this seems like a natural choice. For instance, the field $\mathbb Q_p(\mu_p^{1/p^\infty})$ is studied in ...
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299 views

State of the art on the main conjecture for supersingular elliptic curves/modular forms

Kobayashi formulated the analog of the main conjecture in Iwasawa's theory for elliptic curves which are supersingular at a prime p. This makes use of the $\pm$ Selmer groups which are shown to be ...
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Infinite Non Abelian Extensions Unramified Outside p

Let $K$ be a number field and $p$ be a fixed odd prime. Suppose $\mathfrak{p}\mid p$ is the only prime prime above $p$ in $K$, and that $p$ does not divide the class number of $K$ (I am okay with ...
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$\lambda$-invariants in cyclotomic $\mathbb{Z}_p$ extensions

The idea that Selmer groups and class groups are related is not new. More recently, we understand that the growth patterns of fine Selmer groups are very similar to that of class groups in cyclotomic $...
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126 views

Cassels Pairing for Fine Selmer groups

Let $S_n$ be the Selmer group of $E/K_n$ where $K_n/K$ is the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension of $K$ and $C_n$ be the torsion part of $S_n$. By Cassels pairing, we know that ...
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$\mu=0$ for CM Elliptic curves?

Let $E$ be an elliptic curve defined over $F$ with CM by $\mathcal{O}_K$ where $K$ is an imaginary quadratic field. We may assume the $F$ contains $K$ and also contains the $p$-division points, where $...
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Iwasawa theory and cohomological $p$-dimension of Inertia

Let $E$ be a elliptic curve without complex multiplication over a number field $F$. Let $F_n=F[E_{p^n}]$ and $F_{\infty}=F[E_{p^\infty}]$. So by a well know theorem of Serre, the Galois group $Gal(F_\...
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From which layer we can apply Iwasawa's class number formula in Z_p extension?

Let $K$ be a number field. Let $K_\infty/K$ be a $\mathbb{Z}_p$ extension. Iwasawa proved that there are four integers $n_0,\mu,\lambda,\nu \geq 0$ such that for any $n\geq n_0$, $$\mathrm{ord}_p(...
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$p$-primary torsion of an elliptic curve in the cyclotomic $\mathbb{Z}_p$-extension of a $p$-adic field

Let $K$ be a number field and $v$ be a fixed prime above $p$. Let $k=K_v$. We have the cyclotomic $\mathbb{Z}_p$ extension $K_\infty/K$ and if $w$ is a prime above $v$ in $K_\infty$ we write $k_\infty=...
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Calculating some Galois cohomology

Let $L/\mathbb{Q}$ be a Galois extension of degree $p$ and $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $p$ be a fixed prime (of good ordinary reduction if required). We use $L_\infty, \...
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The Iwasawa $\lambda$-invariant of the cyclotomic $\mathbb{Z}_3$-extension of $\mathbb{Q}(\sqrt{-3})$

I'm now on a research about the Iwasawa $\lambda$-invariants of the cyclotomic $\mathbb{Z}_p$-extensions of number fields. And it happens that the cyclotomic $\mathbb{Z}_3$-extension of $\mathbb{Q}(\...
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Kato's Euler System for Isogenous Elliptic Curves

Let $E,E^\prime$ be elliptic curves over $\mathbb{Q}$ and also suppose they are $p$-isogenous. How are the Euler systems corresponding to the two isogenous elliptic curves related, if at all?
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How does the $\lambda$ invariant propagate with extra ramification?

Let $\mathbb{Q}^{cyc}$ denote the cyclotomic $\mathbb{Z}_p$ extension of $\mathbb{Q}$ and let $\Lambda$ denote the corresponding Iwasawa algebra. Let $p$ be a prime. Let $S$ denote a finite set of ...
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114 views

$\mu=0$ for elliptic curves over number fields

Greenberg conjectured that given $E/K$, there always exists $E^\prime/K$ such that $E'$ is isogenous to $E$ and $\mu(E^\prime)=0$. Michael Drinen has shown that for an elliptic curve $E/K$, it is ...
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241 views

Is there relationship between $\mu=0$ for an elliptic curve and the irreducibility of its residual representation at a prime $p$?

I remember it being mentioned at a talk that if $E$ is an elliptic curve over $\mathbb{Q}$ and $p$ a prime at which it has good reduction then the dual to the Selmer group over the cyclotomic $\mathbb{...
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169 views

How likely is it for Selmer groups to have mu invariant 0?

Given a number field $K$, how likely is it that we'll find at least one elliptic curve $E/K$ such that the $\mu$-invariant of its Selmer group is 0 (in a cyclotomic extension)?
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An issue with showing that an Iwasawa module has zero $\mu$ invariant

Let $\chi$ denote the $p$-adic cyclotomic character. Let $\mathbb{Q}^{cyc}$ denote the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. Let $\gamma$ be the topological generator of $\Gamma=\text{...
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101 views

What is the current status on the corank conjecture for Selmer groups (2)?

This is a follow up to What is the current status on the corank conjecture for Selmer groups? Let E be an elliptic curve over a number field $K$ an imaginary quadratic field in which a prime $p$ ...
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List of applications of the Deformation theory of Galois Representations in contexts other that $R=T$

The Deformation theory of Galois representations developed by Mazur has been extensively used in proving that Galois representations with special properties are modular/automorphic through the ...
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What is the current status on the corank conjecture for Selmer groups?

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ a prime. It is conjectured in the book of Coates and Sujatha "Galois Cohomology of Elliptic Curves" (Conjecture 2.5) that the corank of the ...
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157 views

When can one expect that the $\mu$-invariant of a $\mathbb{Z}_p$-extension of a number field is zero?

What is special about $\mathbb{Z}_p$-extensions which are motivic to ensure that their $\mu$ invariant is zero? Is there a simple conceptual reason. Here are some examples. Let $F$ be a totally real ...
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128 views

Iwasawa Theoretic Interest in a certain type of result

This question is probably going to sound vague (since it does to me) and I wish I could make it more precise, but here goes. For $p\in \{107, 139, 271,379\}$ Ohtani and Blondeau (in separate papers) ...
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Some Questions regarding the fine Shafarevich-Tate group

The fine Shafarevich-Tate (ST) group appears to not have been studied in much depth except for one paper by Wuthrich (2007). However, it only talks about fine ST group for elliptic curves, which makes ...
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174 views

Inverse Problem for Iwasawa Modules

Let $\Lambda$ denote the Iwasawa algebra and $M$ a finitely generated torsion $\Lambda$ module. Does there exist a number field $K$ and a $\mathbb{Z}_p$-extension $K_{\infty}/K$ such that the $p$-...
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47 views

Finite generation for a restricted ramification idele module

Let $k$ be a number field, let $\bar k \subseteq \mathbb{C}$ be a fixed algebraic closure of $k$, and let $S$ be the set of infinite primes of $k$. Denote by $k_S$ the maximal extension of $k$ inside $...
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745 views

Elliptic curves and $GL(2)$ Iwasawa theory

Let $E$ be a elliptic curve without complex multiplication over a number field $F$. Let $F_n=F[E_{p^n}]$ and $F_{\infty}=F[E_{p^\infty}]$. So by a well know theorem of Serre, the Galois group $Gal(F_\...
4
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73 views

non $p$ part of the class group and analogous results

Washington had proven in 1978 that for $q$ a prime ($q \neq p$), if $q^{e_n}$ exactly divides the class number of $\mathbb{Q}_n$, ie the $n$-th layer in the cyclotomic $\mathbb{Z}_p$ extension, then $...
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1answer
147 views

references on group representation over local fields / a question on an argument of a Ralph Greenberg's paper

I'm currently studying Iwasawa theory. 1) There are many $\mathbb{Z}_p$-modules on which some Galois groups act. So I often face some facts on the group representation over local fields or p-adic ...
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173 views

Weak Leopoldt Conjecture for the Split Prime $\mathbb{Z}_p$-extension

In his 1973 Annals paper, Iwasawa proved that the weak Leopoldt Conjecture holds for the cyclotomic $\mathbb{Z}_p$-extension of any number field. If $K$ is an imaginary quadratic field and $F/K$ is ...
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1answer
316 views

A question on a proof in the Ralph Greenberg's paper “On a Certain l-Adic Representation”

I'm currently reading the paper "On a Certain l-Adic Reprersentation" written by Ralph Greenberg.(Inventiones 1973) And I'm stuck with a proof of the Proposition 2. Here $k$ is a totally imaginary ...
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268 views

p-adic L-functions for (dual of) fine Selmer Groups

If $R(E/\Bbb Q_{\infty})$ is the fine Selmer group and $Y(E/\Bbb Q_{\infty})$ is its dual, then we know that $Y(E/\Bbb Q_{\infty})$ is a finitely generated $\Lambda$-module and by a theorem of Kato, ...
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192 views

On an isomorphism between $p$-adic power series and an inverse limit

Let $K$ be an extension field of $\mathbb{Q}_p$, let $O$ be the ring of integers of $K$, and let $P$ be the maximal ideal of $O$. If $K$ is a finite extension of $\mathbb{Q}_p$, there is the well-...
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165 views

Growth of Selmer Groups

If $E$ is an elliptic curve over $K$, is there any effective estimate for the discriminant of the extension $L/K$ for which the $p$-part of the Selmer or Tate-Shafarevich groups become large? I will ...
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120 views

History of the relation between $p$-adic measures and power series

In 1964, Kubota and Leopoldt defined the $p$-adic $L$-function by means of some $p$-adic sums (now called the Volkenborn integral which is a $p$-adic distribution). Later, Mazur (in his secret ...
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173 views

Restricted Iwasawa theory

Let $p$ be a prime number, let $K$ be a number field, and let $L$ be a Galois extension of $K$ such that the Galois group $\Gamma$ of $L$ over $K$ is (continuously) isomorphic to the (additive) group $...
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169 views

Structure of modules over Iwasawa algebra $\mathbb{Z}_p[[T]]$ when taken mod $p$

Let $A \in M_n(\mathbb{Z}_p)$ be a nonsingular matrix which is nilpotent mod $p$, so $A^r \in pM_n(\mathbb{Z}_p)$ for some $r$. Then $\mathbb{Z}_p[[T]]$ acts on $\mathbb{Z}_p^n$ with $T$ acting by $A$....