All Questions
Tagged with class-field-theory arithmetic-geometry
15 questions
15
votes
1
answer
777
views
comparison of completion and Henselization in class field theory
Given a ring $R$ with maximal ideal $\mathfrak{m}$, we can form the localization $R_\mathfrak{m}$, the completion $\hat{R}_\mathfrak{m}$ or the Henselization $\hat{R}^h_\mathfrak{m}$ of $R$ with ...
- 1,338
11
votes
0
answers
381
views
What are the possible bad reductions for an abelian variety of dimension $g$ and a maximal endomorphism ring?
Perhaps the most basic fact about abelian varieties with CM is they have an everywhere potential good reduction (Serre-Tate). On the face of it it might appear that there isn't much more to be added ...
- 13.8k
8
votes
3
answers
1k
views
Deciding a quadratic diophantine equation
Given $a,b\in\Bbb Q_+$, is there an easy way to decide if $$S_{a,b}=\{(x,y)\in\Bbb Z^2:ax^2 + by^2=1\}=\emptyset?$$
I am more interested in seeing if there is a quick way to test for case when ...
- 13.9k
8
votes
1
answer
253
views
Does Ribet's construction of class fields give us eigenspaces of rank 1?
Ribet's paper on the Herbrand-Ribet theorem constructs a representation $\rho: Gal(\overline{\Bbb Q}/\Bbb Q) \to GL_2(\mathbb F_q)$ where $q = p^r$ of the specific form:
$
\begin{bmatrix}
1 & *\\
...
- 7,746
8
votes
1
answer
823
views
Does complex multiplication for higher dimensional abelian varieties give some generalization of class field theory?
I am currently learning some aspects of the theory of complex multiplication for elliptic curves, and the relationship with class field theory.
As I understand it, there is a very special class of ...
- 3,058
5
votes
2
answers
651
views
The $\ell$- part of the class groups of the $p$-cyclotomic fields
Let $K_n = \Bbb Q(\mu_{p^{n+1}})$ and let $A_n$ be it's class group. Iwasawa theory tells us a lot about the $p$-part of $A_n$. For instance, we know quite a lot about how it varies with $n$.
I am ...
- 7,746
4
votes
1
answer
304
views
Topological structure on higher dimensional local fields
Let $F$ be a $n$-dimensional local field. If $n=0$ or $1$, the topological structure on $F$ was well-known, however if $n>1$ i.e, $F$ is a higher dimensional local field, I don't know something ...
- 479
3
votes
0
answers
215
views
Global class field theory and closure of unit groups
I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a ...
- 85
2
votes
0
answers
250
views
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 ...
- 319
2
votes
0
answers
128
views
How do elliptic units generate the module of Euler systems over abelian extensions of imaginary quadratic fields?
I am trying to undesrtand the analogy between the Euler systems over abelian extensions of the rationals and the Euler systems over abelian extensions of imaginary quadratic fields.
As Soogil Seo ...
- 99
2
votes
0
answers
254
views
Global sections of higher direct images
If $f : X\to V$ is a smooth proper map of smooth schemes, what are the global sections of
$R^if_{fppf, *}\mu_p$
$R^if_{fppf, *}\mathbb{G}_{\rm m}$
I was reading Milne's book "Arithmetic duality", ...
user113393
1
vote
1
answer
340
views
Brauer group of global fields
Is the Brauer group $\text{Br}(K)$ of a global field $K$
an $\ell$-divisible group for some prime $\ell$? If so, what $\ell$?
Is $\text{Br}(K)[n]$ finite, for $n$ integer?
I know from class field ...
user120812
1
vote
0
answers
210
views
What is a definition of $A(P_v)$ in the definition of Brauer-Manin obstruction?
This is a question related to the definition of Brauer-Manin obstruction.
Let $K$ be a number field. $X/K$ be an algebraic variety over $K$.
Let $O_{X,P}$ be a local ring of $X$ at $P$. Let $Br(X)=\...
- 1,541
1
vote
0
answers
180
views
Maximal unramified extension and algebraic closure of $\operatorname{Frac}(\widehat{A_{\mathfrak{m}_A}})$
$\DeclareMathOperator\trdeg{trdeg} \DeclareMathOperator\Frac{Frac} $
Let $k$ be an algebraically closed field of characteristic $0$ and $K$ a function field over $k$. let $(A, \mathfrak{m}_A)$ a ...
- 5,966
1
vote
0
answers
54
views
Finite generation for a restricted ramification idele module
Let $k$ be a number field, let $\bar k \subseteq \mathbb{C}$ be a fixed algebraic closure of $k$, and let $S$ be the set of infinite primes of $k$. Denote by $k_S$ the maximal extension of $k$ inside $...
- 11.3k