19 votes
Accepted

Why is the regulator of the degree 11 extension of $\mathbb{Q}$ so large?

A prime $\ell$ splits in $K_p$ for $p$ odd if and only if $$\ell^{p-1} \equiv 1 \mod p^2.$$ We can express $\lim_{s \to 1} (s-1) \zeta_{K_n(s)}$ as the product of $\frac{1}{ 1- |\mathfrak a|^{-1} }$ ...
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  • 116k
13 votes
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Why can Euler systems constructed from algebraic cycles only be anticyclotomic?

Let me explain a bit more what that footnote was supposed to mean. As I'm sure you know, an Euler system for a Galois representation $V$ over a number field $K$ consists of a bunch of classes in $H^1(...
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9 votes
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Elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$

EDIT: There was a problem with my original answer. Details below the bottom line. If you have any elliptic curve $E/{\mathbb Q}$ with a point of order 3, then we have an exact sequence of Galois ...
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  • 1,650
9 votes
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Finding out $p$-torsion elements of an elliptic curve $E$ over $\mathbb{Q}_p$

Suppose $p>2$ and that $E$ has good reduction. If the reduction $\tilde E(\mathbb{F}_p)$ has no $p$-torsion then there is no $p$-torsion in $E(\mathbb{Q}_p)$. Otherwise look at the exact sequence $$...
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7 votes
Accepted

Value of the quadratic Gauss sum viewed in $\mathbb{C}_p$

That $G(\psi,\zeta)^2 = \psi(-1)p$ is pure algebra, so it holds in $\mathbf C$ or $\mathbf C_p$ or any other field not of characteristic $2$ that contains a nontrivial $p$th root of unity. You could ...
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  • 41.8k
6 votes
Accepted

On an isomorphism between $p$-adic power series and an inverse limit

There is unique division with remainder by a monic polynomial in $O[X]$, where $O$ is any commutative ring. When $O$ is a $p$-adically complete ring, the Weierstrass division theorem tells us there is ...
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  • 41.8k
6 votes
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The $\ell$- part of the class groups of the $p$-cyclotomic fields

As Keith Conrad observed in his comment, what you are asking about is Washington's theorem which is either in his book, referenced as Theorem 16.12, or in his original paper in Invent. Math. He proved ...
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6 votes
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Studying a connection between Iwasawa theory and the $K(n)$-local Picard group by some geometry on the Lubin-Tate stack

A long time ago there were various attempts to make this sort of approach work, but they were not very successful. If you want to try again then you should make sure that you are familiar with the ...
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6 votes
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Describing the Gamma-transform explicitly in terms of power series

This is a hard problem (and one which is easily overlooked by the unwary)! Just to be clear, I'll summarize (how I think about) the problem: as a relation between additive and multiplicative Fourier ...
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5 votes
Accepted

What is the current status on the corank conjecture for Selmer groups?

Yes, the corank conjecture is a theorem for elliptic curves over $\mathbb{Q}$. The key to the proof is the following: Theorem (Kato, 2004): For any $E$ and any $p$, the "fine Selmer group" $Sel_p^0(...
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5 votes
Accepted

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

Yes, the main conjecture is isogeny-invariant. See here: B. Perrin-Riou, Variation de la fonction $L$ $p$-adique par isogénie, Algebraic number theory, Adv. Stud. Pure Math. 17 (1989), pp. 347-358.
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5 votes
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Refinement of (classical) Iwasawa main conjecture

There is an answer which follows from Kolyvagin's theory of Euler Systems, and can be found in Theorem 4.4 of K. Rubin, Kolyvagin's System of Gauss Sums, in Arithmetic Algebraic Geometry, van der ...
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5 votes

Classification of cyclotomic fields with class number 1

The complete list of $n$ for which ${\bf Q}(\zeta_n)$ has unique factorization is 1 through 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 54, 60, 66, 70, 84, 90. Source; ...
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4 votes
Accepted

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

Yes. If $E$ has potentially good reduction, this is due to H. Imai, Proc Japan Adac Math Sci 51 (1975). A non-standard proof is Theorem A.2.8 in Coates-Sujatha's "Galois cohomology of elliptic curves"
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4 votes
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Some questions related to Iwasawa invariants of elliptic curves

1) Iwasawa theory, as practiced by K.Iwasawa, is concerned with $\mathbb Z_{p}$-extensions. There is only one $\mathbb Z_{p}$-extension of $\mathbb Q$. Over more generally number fields, and in more ...
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  • 9,500
4 votes

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

The full main conjecture is invariant under isogenies defined over $\mathbb{Q}$, not just the statement about $\mu$ and $\lambda$-invariants. The only thing that changes on the analytic side is the ...
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4 votes
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Elliptic curves and $GL(2)$ Iwasawa theory

For $n$ large enough the Galois group of $F_n/F_{n-1}$ identifies with the kernel $G$ of $\operatorname{GL}_2(\mathbb{Z}/p^{n+1}\mathbb{Z}) \to \operatorname{GL}_2(\mathbb{Z}/p^{n}\mathbb{Z})$. Let $P$...
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4 votes

Herbrand-Ribet and Mazur-Wiles for function fields

(I was hoping somebody else would answer this, because function fields are not really my area and I hoped I would learn something from the answer; but nobody seems to be biting, so...) Iwasawa theory ...
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4 votes

Kato's Euler System for Isogenous Elliptic Curves

They are related because they come from the same element for $X_1(N)$. Suppose $E$ is the one of the two with the smaller degree of the modular parametrisattion $X_1(N)\to E$ of minimal degree. Then ...
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4 votes
Accepted

Is there relationship between $\mu=0$ for an elliptic curve and the irreducibility of its residual representation at a prime $p$?

I think generalizing a conjecture we know sol little about is a risky business, but let me try to say something non-vacuous. First of all, I'm assuming that $E$ has good ordinary reduction (...
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  • 9,500
4 votes

Iwasawa theory and perfectoid spaces

A flippant response is that people had the idea of using perfectoid theory in Iwasawa theory long before perfectoid theory even existed. What I'm referring to here is the work of Fontaine--...
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4 votes
Accepted

Modular symbols associated to Rankin Selberg convolutions and the symmetric square

I do not think there is a reference for this theory, because as far as I know no such theory exists. I have spent a substantial portion of my career studying the arithmetic of the special values of ...
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4 votes

Completed cohomology and variants

There is no reason why you shouldn't consider completed etale cohomology, instead of completed singular cohomology. If you look at Emerton's 2006 Inventiones paper which started the whole theory, he ...
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4 votes

Classification of finitely generated modules over non-commutative rings

I haven't thought about base changes, but the original problem for $\alpha=1$ (so $\sigma$ is the identity map and $R=\mathbb{Z}_p[[t]][F]$ is just a polynomial ring over $\mathbb{Z}_p[[t]]$, and ...
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4 votes

Value of the quadratic Gauss sum viewed in $\mathbb{C}_p$

$\DeclareMathOperator\sgn{sgn}$Just as in the complex case, we have that $G(\sgn, \chi)$ lies in $\mu_4(\mathbb C_p)\sqrt q$. We have that $G(\sgn, \chi)$ equals $\sum_{c \in \mathbb F_q} \chi(c^2)$, ...
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  • 7,923
4 votes

Order of $37$-Sylow subgroup of ideal class group of $K_{37} = \Bbb Q(\mu_{37^{n}})$ is known to be $37^n$

Much more than Iwasawa's original theorem is known by now. First of all, the $p$-primary part of the class group stays trivial in the field $K_n=\mathbb{Q}(\mu_{p^{n+1}})$ unless it is already non-...
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4 votes
Accepted

A confusion about power series and p-adic measures

You are asking about substitution of one power series into another. This can be interpreted in two ways, which ultimately amount to the same thing (like two different ways of thinking about anything) ...
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  • 41.8k
3 votes
Accepted

State of the art on the main conjecture for supersingular elliptic curves/modular forms

Theorem (Xin Wan): If $E$ is an elliptic curve of square-free conductor $N$, and $p \ge 3$ is a prime such that $p \nmid N$ and $a_p(E) = 0$, then Kobayashi's $\pm$ Iwasawa main conjectures are true ...
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3 votes
Accepted

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

I don't think we know how to prove this directly. Indeed, recent works by Wake and Wake–Erickson show that this cyclicity is equivalent to a conjectured improvement of Mazur–Wiles' result to the ...
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3 votes
Accepted

Completely split primes in non-anticyclotomic $\mathbb{Z}_p$-extensions

The article Sur les ideaux dont l'image par l'application d'Artin dans une $\mathbb{Z}_p$-extension est triviale of Michel Emsalem provides a satisfactory answer to the general question of how many ...
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