The iwasawa-theory tag has no usage guidance.

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### characteristic ideal of the Iwasawa module

Let $H$ be a complex biquadratic Galois extension of $\mathbb{Q}$ such that the galois group of $H$ is isomorphic to the Klein Group. Let $H_{\infty}$ be an anticyclotomic $\mathbb{Z}_p$-extension of $...

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### Finding out $p$-torsion elements of an elliptic curve $E$ over $\mathbb{Q}_p$

Let $E$ be an elliptic curve over $\mathbb{Q}$.
Then how to compute the $p$-torsion elements of $E$ over the $p$-adic field $\mathbb{Q}_p$ using SAGE or any other means ?
At least can we say whether ...

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### Iwasawa lambda Invariant for CM type fields

In the following I will follow the notations as in Chapter $11$, section $3$ of the book 'Cohomology of number fields' by Neukirch and others. Let $k_{\infty}$ be a $\mathbb{Z}_p$-extension of a ...

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### Finiteness of the $p$-primary subgroup of an elliptic curve over the cyclotomic $\mathbb{Z}_p$-extension

Let $E$ be an elliptic curve defined over a number field $F$ and $F_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $F$. Is it true that the $p$-primary subgroup of $E$ over $F_\infty$ i.e. $E[p^...

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### Iwasawa theory, $\mathbb{Z}_p^{2}$-extension, Greenberg module

Take $H\subset \bar{\mathbb{Q}}$ be a quartic imaginary number field such that $\operatorname{Gal}(H/\mathbb{Q})=\mathbb{Z}_2 \times \mathbb{Z}_2$. Denote by $F$ the quadratic real subfield of $H$ and ...

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### How to compute group homology of Iwasawa algebra

Let $G$ be a $p$-adic Lie group, $H$ a subgroup of $G$. What is $H_1(H,\Lambda(G))$, where $\Lambda(G)$ is the Iwasawa algebra of $G$ over $\mathbb Z_p$?
If it simplies the question, we may assume $G$...

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### Greenberg and Iwasawa Theory

Let $p$ be a prime number and $H$ be number field. Denote by $\tilde{H}$ the composite of all $\mathbb{Z}_p$ extensions of $H$, so $\operatorname{Gal}(\tilde{H}/H)=\mathbb{Z}_p^{d}$. Write $\varLambda$...

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### Torsionfree finitely generated compact Iwasawa module

The following fact falls under the category of Iwasawa modules.
Let $M$ be a torsion free finitely generated module over the non commutative noetherian ring $\Bbb{Z}_p[[G]]$, (where $G$ is a p ...

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### Iwasawa's mu-invariant for noncyclotomic $\mathbf{Z}_p$ extensions of cyclotomic fields?

Let $p$ be an odd prime number, $m$ a positive integer with $p\mid m$. Put $k=\mathbf{Q}(\mu_m)$.
(1) Is there any example where certain noncyclotomic $\mathbf{Z}_p$-extension $k_\infty/k$ has ...

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### Iwasawa theory: Do these $\mu$-invariants of a number field coincide?

Let $k$ be a number field and let $k_\infty$ be the cyclotomic $\mathbf{Z}_p$-extension of $k$. Put $\Gamma=G(k_\infty/k)\cong \mathbf{Z}_p$, $\Lambda=\mathbf{Z}_p[[\Gamma]]$. Let $S$ be a finite set ...

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### Some questions related to Iwasawa invariants of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$.
Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...

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### Main conjecture for elliptic curves invariant under a $\mathbb{Q}$-isogeny

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ \lambda_{E}^...

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### Main conjecture for elliptic curves

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ \lambda_{E}^...

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### Elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$

How to find out examples over elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$ $?$

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### Splitting as $\mathbb{F}_p[[X]]$-modules

Let $A$ be a finitely generated torsion $\mathbb{Z}_p[[X]]$-module, $B$ = { $x \in A$ such that $px=0$ } and $C=A/B$ where $\mathbb{Z}_p$ denotes the $p$-adic integers. Given $ 0 \rightarrow B/pB \...

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### $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2

Consider the elliptic curves -
$ E_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $
$ E_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $
with both good ...

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### $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 1

1) What are the examples of elliptic curves over $\mathbb{Q}$ with good reduction and $\mu$-invariant $\geq 2$ at $p = 3$ and how to find them $?$
2) Let $\Lambda = \mathbb{Z}_{p}[[T]] $ and $ K=\...

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### Iwasawa's invariants

I want a reference that treated the proof of this proposition :
Proposition : Suppose that $X$ is a finitely generated, torsion $\Lambda-$module. Then there are uniquely determined $\Lambda-$...

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### $\lambda$-invariant is constant for isogenous elliptic curves

How to prove that the $\lambda$-invariant is constant for isogenous elliptic curves $?$

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### Periods for 2-variable p-adic L-functions

Hi all,
I am sorry to ask a stupid question but I am really confused right now. Kitagawa-Mazur constructed a $2$-variable p-adic L-function attached to Hida families of modular forms. For their ...

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### Applications of Iwasawa Theory

Iwasawa theory gives a formula for the power of $p$ dividing the class group of the $\mathbb{Q}(\zeta_{p^n})$ (where $\zeta_{p^n}$ is a primitive root of unity of exact order $p^n$) for sufficiently ...

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### Where in the literature does the anticyclotomic $\mathbf{Z}_p$-extension of an imaginary quadratic field first appear?

If $K$ is an imaginary quadratic field, then the $\mathbf{Z}_p$-rank of $K$ is $2$, meaning that the Galois group of the compositum of all the $\mathbf{Z}_p$-extensions of $K$ in an algebraic closure $...

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### main conjecture of Iwasawa theory implies Herbrand-Ribet

I need a reference/proof for "main conjecture of Iwasawa theory => refinement of Herbrand-Ribet ($v_p(B_{p-i}) = v_p(|A_i|)$, where $A_i$ denotes the $i$-th eigenspace of the Galois group acting on ...

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### Vanishing of certain $\mu$-invariants attached to abelian extensions of imaginary quadratic fields

In Fonctions L p-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes,
J. Reine Angew. Math. 358 (1985), 76–91, Roland Gillard shows the following result (I mainly follow the ...

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### Geometric interpretation of Hida isomorphism

[EDIT]: After getting a very nice answer by Kevin Buzzard I realize that my question was a little bit too vague and I try to restate it more precisely.
As the title says, I would like to understand ...

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### Prime Decomposition in Cyclotomic Z_p-extensions

In their classic paper "Class fields of abelian extensions of $\mathbf{Q}$", Mazur and Wiles assert that
"in a cyclotomic
$\mathbf{Z}_p$-extension only finitely
many primes lie above any ...

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### Gouvea-Mazur conjecture

I was reading about the conjecture made by Gouvea and Mazur in their paper "Families of modular eigenforms" which says that if $k_1 \equiv k_2 \pmod {p^{n}(p-1)}$ for some integer $n\geq \alpha$. then ...

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### Two-variable p-adic L-functions of elliptic curves

Suppose $K$ is an imaginary quadratic field (with class number 1, for simplicity), $p \ne 2$ a prime split in $K$, and $K_\infty$ the $\mathbb{Z}_p^2$-extension of $K$.
If $E / \mathbb{Q}$ is an ...

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### Functional equations relating to p-adic L-functions

Let f be a modular form of weight k for $\Gamma_0(N)$. Let us assume that $p\not\vert$N. Then we can construct 2 p-adic L-functions corresponding to the 2 roots $\alpha$ and $\beta$ of the equation $x^...

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### rational points on algebraic curves over $Q^{ab}$

Motivation:
Let $Q_{\infty,p}$ be the field obtained by adjoining to $Q$ all $p$-power roots of unity for a prime $p$. The union of these fields for all primes is the maximal cyclotomic extension $Q^...

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### Non-vanishing of p-adic L-functions

In Non-vanishing of L-series of modular forms (easy case?) it was answered that for a cuspidal newform $f$ of weight strictly greater than 2, then $L(f,1)$ is non-zero. (Here the $L$-series is ...

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### Iwasawa mu-invariant for abelian extensions of quadratic number fields

Let K be a number field and $p$ an odd prime. Let $\mu$ be the Iwasawa $\mu$-invariant of the class group of the cyclotomic $\mathbb{Z}_p$-extension of $K$. If $K$ is abelian over $\mathbb{Q}$ then it ...

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### Fontaine's classification of p-divisible groups

Let k be a finite extension of $\mathbb{F}_p$, and W its ring of Witt vectors. Write W[F,V] for the Dieudonn\'e ring.
Let G be a connected p-divisible group which is finite-dimensional over k, and ...

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### Iwasawa main conjectures vs Bloch-Kato conjectures

Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the ...

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### Does the everywhere unramified extension of Q(mu_37) of degree 37 grow into a Z_37-extension?

Let $p=37$. Since $p$ divides the numerator of $B_{32}$, by Ribet's proof of the converse of Herbrand's theorem, we know that the class group of ${\bf Q}(\mu_p)$ has size divisible by $p$. More ...

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### Are Kato's zeta elements integral?

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $T$ the $p$-adic Tate module of $E$. Kato's Euler system, constructed in the paper "P-adic Hodge theory and values of zeta functions of modular forms"...

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### P-adic L functions

My question is how should one think of p-adic L functions? I know they have been constructed classically by interpolating values of complex L-functions. Recently I have seen people think about them in ...

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### Characteristic Complexes in Iwasawa theory

For all that follows, $p$ is a fixed odd prime. In the formulation of the Noncommutative Main Conjecture of Iwasawa theory one uses étale cohomology to define an algebraic object analogous to Iwasawas ...

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### Capitulation in cyclotomic extensions

Let $p$ be an irregular prime, which means that $p$ divides some Bernoulli number: $p \mid B_k$ (for some even $k\in[2,p-3]$). This implies that the class number of the field $K$ of $p$-th roots of ...

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### A question about Iwasawa Theory

I am just reading about Iwasawa theory about Coates and Sujatha's book on Iwasawa Theory. I was wondering that since Iwasawa thought about the whole theory from the analogy of curves over finite ...

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### P-adic L-functions of nonabelian twists of elliptic curves

Let $E$ be an elliptic curve and $\rho$ an Artin representation of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. Then there is a "twisted L-function" $L(E, \rho, s)$, corresponding to the ...

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### non-commutative iwasawa theory

In commutative Iwasawa theory, the main conjecture states that the p-adic L-function generates the characteristic ideal of an algebraic object. Non-commutative Iwasawa theory seems to mimik this - ...

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### Existence of multi-variable p-adic L-functions

What's the "state of the art" in constructing multi-variable p-adic L-functions for number fields?
More precisely: if K is a number field, and $K_{\infty} / K$ is an infinite Galois extension, ...

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### When does a p-adic function have a Mahler expansion?

Let $f: \mathbb{Z}_p \rightarrow \mathbb{C}_p$ be any continuous function. Then Mahler showed there are coefficients $a_n \in \mathbb{C}_p$ with
$$
f(x) = \sum^{\infty}_{n=0} a_n {x \choose n}.
$$
...

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### Special values of $p$-adic $L$-functions.

This is a very naive question really, and perhaps the answer is well-known. In other words, WARNING: a non-expert writes.
My understanding is that nowadays there are conjectures which essentially ...

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### von Staudt-Clausen over a totally real field

Before I ask the question, I need to recall what Bernoulli numbers
$(B_k)_{k\in\mathbb{N}}$ are, and what von Staudt and Clausen discovered about
them in 1840. The numbers $B_k\in\mathbb{Q}$ are the ...

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### Does p-adic $L$- function determine the $L$ function

Let $E_1$ and $E_2$ be two Elliptic curve defined over $\mathbb Q$ . Let $p$ be an fixed given odd prime of $\mathbb Q$ at which both the curves have good ordinary reduction. Moreover p-adic $L$-...

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### Model theoretic applications to algebra and number theory(Iwasawa Theory)

One of my favorite results in algebraic geometry is a classical result of AX (see http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/) I'll recall ...

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### How do we study Iwasawa theory?

What papers should we read to start? What basic knowledge do we need to understand the question? What is this area really about? And what are people researching on it? Thank you!