19
votes
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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 $\frac{1}{ 1- |\mathfrak a|^{-1} }$ ...
- 148k
15
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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13
votes
Accepted
Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why?
Symplectic case: Here are two reasons (not necessarily the only ones) why $\operatorname{GSp}_{2n}$ is more convenient to work with than $\operatorname{Sp}_{2n}$.
Firstly: there is no Shimura datum ...
- 37k
9
votes
Accepted
Why $p$-adic measures?
This is extremely late, but hopefully it's still of some use/interest to you.
p-adic L-functions are usually described as 'p-adic interpolations of special values of L-functions'. For Kubota--Leopoldt'...
- 206
7
votes
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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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7
votes
How is Taylor-Wiles patching "horizontal Iwasawa theory"?
I think your question already contains its own answer.
In classical, "vertical" Iwasawa theory one studies class groups, or other arithetic widgets like elliptic curve Selmer groups, in a ...
- 37k
7
votes
Accepted
Is there something I am missing about the computation of the $p$-part of the class groups of cyclotomic fields?
[Rather than leaving the comment "Class number formula" for Olivier as a comment, I expand it for other readers of the question, to a partial answer.]
Kummer knew in 1850 that the class ...
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6
votes
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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, meaning the natural ring homomorphism $O \to \...
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6
votes
Accepted
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 ...
- 5,670
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 ...
- 56.9k
6
votes
Accepted
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 ...
- 37k
6
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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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 ...
- 5,670
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
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 (...
- 10.9k
5
votes
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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 ...
- 111
5
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 ...
- 37k
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; ...
- 51
5
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-...
- 8,945
5
votes
Is there something I am missing about the computation of the $p$-part of the class groups of cyclotomic fields?
Recently, I stumbled coincidentally on the paper
Computation of invariants in the theory of cyclotomic fields K. Iwasawa and C. Sims J. Math. Soc. Japan Vol.18 (1966)
This explains in full details how ...
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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 ...
- 37k
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 ...
- 35.2k
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 ...
- 8,945
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--...
- 37k
4
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 ...
- 37k
4
votes
Accepted
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 $P$...
- 8,945
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 ...
- 37k
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 $\sum_{c \in \mathbb F_q} \chi(c^2)$, ...
- 12.9k
4
votes
Accepted
What's the class group of $\mathbb{Q}^{\mathrm{ab}}$?
From Armand Brumer, The class group of all cyclotomic integers:
As an abelian group, $\mathrm{Pic}(O_\infty)$ is isomorphic to a countable direct sum of copies of $\mathbb Q/\mathbb Z$.
Here $O_\...
- 28.2k
4
votes
Accepted
Integration against Eisenstein series can be regarded as a cup product
Yes, that does indeed sound like something I might have said :)
I was referring to some extremely powerful theorems, originally due to Michael Harris, which show that:
The cohomology groups of ...
- 37k
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