Questions tagged [iwasawa-theory]
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64
questions with no upvoted or accepted answers
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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 ...
9
votes
0
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744
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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, ...
7
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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 $...
7
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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 ...
6
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answer
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How hard is it to find the first layer of this basic $\mathbb{Z}_p$-extension?
$\DeclareMathOperator\Gal{Gal}$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 $\mathbb{Q}(\zeta_{p^2})$ ...
6
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279
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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 ...
6
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210
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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 ...
6
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291
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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 ...
5
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278
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Why does passing to a $\mathbf{Z}_p$-extension make things easier?
In Iwasawa theory, even if one is only interested in questions about a number field $K$ (e.g. class groups of $\mathbf{Q}(\mu_p)$, Selmer groups of abelian varieties over $\mathbf{Q}$), to prove deep ...
5
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90
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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 ...
5
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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 ...
5
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241
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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 $...
5
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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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255
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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 ...
5
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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 $...
5
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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 ...
4
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Alternative formulation of the Ferrero-Washington Theorem
The Ferrero-Washington theorem says that if $K/\mathbf{Q}$ is an abelian extension, then the cyclotomic $\mathbf{Z}_p$ extension $K^{\text{cyc}}/K$ has $\mu=0$.
In the paper "Iwasawa invariants ...
4
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The Gamma-transform and $p$-adic $L$-functions
I'm currently reading the paper "On the $\mu$-invariant of the $\Gamma$-transform of a rational function" by W Sinnott. In this paper, he gives an alternate proof that $\mu=0$ for abelian ...
4
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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}$...
4
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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, \...
4
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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
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189
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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 $...
4
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209
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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
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202
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Generalisation of Sharifi's conjecture for Siegel varieties
I recently learned Sharifi's conjecture in this article by F.Takako and K.Kato.
According to my understanding, this conjecture states that (the conjugation invariant) of the projective limit $\...
3
votes
0
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178
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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 ...
3
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129
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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 ...
3
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0
answers
149
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Relationship between fine Selmer groups and class groups
Given an elliptic curve $E_{/\mathbb{Q}}$ and an odd prime number $p$, let $S$ be the set of primes consisting of $p$ and the primes at which $E$ has bad reduction. The fine Selmer group $R_{p^\infty}(...
3
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139
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What do congruences between modular forms tell us about $\mu$-invariants of elliptic curves?
This question is based off these notes by Preston Wake about Iwasawa invariants and Hida Families. In the notes, the author asks "why" the elliptic curve $11A3$ has $\mu$-invariant equal to $...
3
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152
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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 ...
3
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107
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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 ...
3
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145
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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_\...
3
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61
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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 ...
3
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126
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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 ...
3
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241
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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 $...
3
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147
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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 ...
3
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122
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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$...
2
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1
answer
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On Iwasawa theory of elliptic curves in $\mathrm{PGL}_2(\mathbb{Z}_p)$-extension
Let $E$ be an elliptic curve over the rationals $\mathbb{Q}$. We consider the Galois representation attached to $E$ by acting on its $p$-adic Tate module $T_p(E)$,
$$
\rho_E: G_{K} \rightarrow \mathrm{...
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Maximal p-extension and pro-p extension
I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help.
Q_1: About terminology $p$-extension.
I find many reference use maximal $p$-extension or maximal abelian p-extension ...
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Compositum of field extensions in context of $\mathbb Z_p$ extension
I had asked this question on stackexchange and I think it is better suited for this site.
Suppose I have a $\Gamma \simeq \mathbb Z_p $ extension $F_\infty /F$ of a number field $F$. Let $F_n$ be the ...
2
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Question about infinitude of $m$-irregular primes
Let $p>2$ be a prime, $\zeta$ a primitive $p$th root of unity, and $m >0$ a square-free integer such that $m \not\equiv 3 \mod 4$ and $\gcd(m,p)=1$. Let $\chi$ be the imaginary quadratic ...
2
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147
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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 ...
2
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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 ...
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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
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255
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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 $N(d)$ ...
2
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201
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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 ...
2
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91
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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?
2
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163
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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=...
2
votes
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81
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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{...
2
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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 $p$ ...
2
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0
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132
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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) ...