# Questions tagged [iwasawa-theory]

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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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### 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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### 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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### 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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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 $\mathbb{Q}(\zeta_{p^2})$ such that $[\mathbb{Q}_1:\mathbb{... • 649 3 votes 0 answers 115 views ### Average size of class groups of cyclotomic fields: three perspectives Let$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}$... • 21.7k 2 votes 1 answer 120 views ### Existence of non-zero pseudo-null submodules Let$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$-... 2 votes 0 answers 105 views ### Iwasawa theory over function fields - How do eigenvalues vary in$\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$on ... • 7,062 3 votes 0 answers 141 views ### Finiteness of points over the cyclotomic extension for modular forms Let$\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$be the vector ... • 303 2 votes 0 answers 90 views ### Iwasawa's results about relation between Galois cohomology and principal factorization Let$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)$to principal factorization in K? I'd try to find ... 2 votes 0 answers 173 views ### Merel's theorem on uniform bound for torsion of all elliptic curves I am reading Silverman's book on Arithmetic of elliptic curves. There he mentions a theorem of Merel (Thm 7.5.1) which reads like this. Thm: For every integer$d \geq 1$, there is a constant$N(d)$... • 303 2 votes 0 answers 145 views ### Tate module of elliptic curves; Commuting Hom functor and tensor product in the second coordinate Let$\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}$be its Pontryagin dual (i.e the ... • 303 2 votes 0 answers 81 views ### Elementary Iwasawa module Let$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$-module? • 21 5 votes 1 answer 252 views ### Classification of finitely generated modules over non-commutative rings Let$\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\}$... • 7,062 4 votes 1 answer 131 views ### Completely split primes in non-anticyclotomic$\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$... 2 votes 0 answers 78 views ### Regular growth of ranks in Iwasawa tower$\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$is a non-commutative torsion ... • 303 1 vote 1 answer 235 views ### Completed cohomology and variants I am interested in the following set up: I have an ind-sequence of curves$\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$cover ... • 7,062 8 votes 1 answer 479 views ### Why can Euler systems constructed from algebraic cycles only be anticyclotomic? In a footnote to the 2018 Zerbes-Loeffler lecture notes from the Arizona Winter School, it's stated that Euler systems constructed from algebraic cycles "cannot give a full Euler system, only an ... • 1,550 5 votes 1 answer 251 views ### Adjoint Selmer groups and Deformation rings Let$\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$-ordinary Hecke eigencuspform, let$... 210 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 ... 251 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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### 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 $\mathbb{Z}_p$ ... 129 views

### 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 ... 1 vote
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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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### 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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### 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 ... 1 vote
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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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