# Tagged Questions

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

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