# Questions tagged [iwasawa-theory]

The iwasawa-theory tag has no usage guidance.

**5**

votes

**1**answer

119 views

### 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?

**2**

votes

**0**answers

45 views

### 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 ...

**1**

vote

**0**answers

102 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 ...

**3**

votes

**1**answer

156 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{...

**2**

votes

**1**answer

118 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)?

**2**

votes

**0**answers

53 views

### 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{...

**2**

votes

**0**answers

83 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 $p$ ...

**5**

votes

**0**answers

106 views

### 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 ...

**5**

votes

**1**answer

262 views

### 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 ...

**4**

votes

**0**answers

109 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 ...

**1**

vote

**0**answers

83 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) ...

**4**

votes

**0**answers

71 views

### 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 ...

**2**

votes

**1**answer

152 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 $p$-...

**1**

vote

**0**answers

42 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 $...

**5**

votes

**1**answer

253 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_\...

**5**

votes

**0**answers

64 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 $...

**1**

vote

**1**answer

139 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 ...

**3**

votes

**0**answers

122 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 ...

**1**

vote

**1**answer

283 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 ...

**6**

votes

**1**answer

203 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, ...

**2**

votes

**1**answer

144 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-...

**2**

votes

**0**answers

154 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 ...

**4**

votes

**0**answers

86 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 ...

**4**

votes

**0**answers

155 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 $...

**5**

votes

**0**answers

136 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 $A$....

**3**

votes

**0**answers

94 views

### For Hida theory on $GU(2,2)$ can $p$ be inert in the imaginary quadratic field $K$?

I am familiar with the theory of Hida families of modular forms, so Hida theory on $GL_2$, but I am not familiar with Hida theory on any other algebraic groups. My question concerns Hida families of ...

**6**

votes

**1**answer

231 views

### Herbrand-Ribet and Mazur-Wiles for function fields

Is there a version of Herbrand-Ribet or Mazur-Wiles (relating divisibility of class groups to special values of L-functions) for functions fields (over finite fields)?
Probably the proofs would have ...

**5**

votes

**1**answer

265 views

### Rationality of trace of endomorphism of Iwasawa-thing

Let $n$ be a positive integer, and $p$ a prime number. Let $K_i$ be the cyclotomic field containing exactly the $np^i$th roots of unity. Let $H$ be the inverse limit of $p$-power torsion of the class ...

**4**

votes

**1**answer

169 views

### Refinement of (classical) Iwasawa main conjecture

Let $p$ be an odd prime, and denote by $Cl_p(H)$ the $p$-part of the ideal class group of a number field $H$. Let $\Delta:=Gal(\mathbb{Q}(\mu_p)/\mathbb{Q})$ and $\omega : \Delta \longrightarrow \...

**1**

vote

**0**answers

111 views

### Motivation for the definition of push-out for $G$-torsors (as seen in Fukaya-Kato)

In the introductory sections to their paper "A Formulation of Conjectures on $p$-adic Zeta Functions in Non-commutative Iwasawa Theory," Fukaya and Kato describe an explicit construction of ...

**8**

votes

**1**answer

173 views

### Does Ribet's construction of class fields give us eigenspaces of rank 1?

Ribet's paper on the Herbrand-Ribet theorem constructs a representation $\rho: Gal(\overline{\Bbb Q}/\Bbb Q) \to GL_2(\mathbb F_q)$ where $q = p^r$ of the specific form:
$
\begin{bmatrix}
1 & *\\
...

**14**

votes

**0**answers

512 views

### How to approach the Mazur-Wiles paper on Iwasawa theory?

I would like to read and understand the Mazur-Wiles paper on Iwasawa theory: "Class Fields of Abelian Extensions of $\Bbb Q$". What would be the right way to approach this paper?
Currently, my ...

**4**

votes

**2**answers

228 views

### The $\ell$- part of the class groups of the $p$-cyclotomic fields

Let $K_n = \Bbb Q(\mu_{p^{n+1}})$ and let $A_n$ be it's class group. Iwasawa theory tells us a lot about the $p$-part of $A_n$. For instance, we know quite a lot about how it varies with $n$.
I am ...

**5**

votes

**0**answers

190 views

### Derivatives of p-adic L-functions of modular forms

Let $f$ be a eigen-newform and $p$ is a good prime for $f$. We know that the $p$-adic $L$-function of $f$ interpolates the complex $L$-values of $f$ when evaluated at Dirichlet characters.
My ...

**4**

votes

**0**answers

113 views

### Structure theorem for modules over multi-variable Iwasawa algebras

It is well-known that if $\Lambda=Z_p[[X]]$ and $M$ a finitely generated $\Lambda$-module, then $M$ is pseudo-isomorphic to
$$
\Lambda^{\oplus r}\oplus\bigoplus_{i=1}^s\Lambda/(F_i)
$$
for some ...

**3**

votes

**0**answers

169 views

### 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 $...

**1**

vote

**1**answer

140 views

### 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 ...

**3**

votes

**0**answers

127 views

### 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 ...

**4**

votes

**1**answer

99 views

### 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^...

**6**

votes

**0**answers

253 views

### 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 ...

**3**

votes

**0**answers

94 views

### 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 $G$...

**2**

votes

**1**answer

521 views

### 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 $\varLambda$...

**2**

votes

**0**answers

96 views

### 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 ...

**6**

votes

**2**answers

383 views

### 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 ...

**5**

votes

**1**answer

205 views

### 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 ...

**2**

votes

**1**answer

318 views

### 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 ...

**4**

votes

**2**answers

242 views

### 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}^...

**7**

votes

**1**answer

993 views

### 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}^...

**5**

votes

**2**answers

341 views

### 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 $p=3$ $?$

**3**

votes

**1**answer

313 views

### 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 \...