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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 ...
Pierre MATSUMI's user avatar
Pierre MATSUMI
  • 781
3 votes
1 answer
322 views

Where am I going wrong in this interpretation of 1-dimensional geometric class field theory?

I posted this on MSE, but didn't get any responses, so I'm reposting here. I tried to write down an example of the main theorem of geometric class field theory, but I must be misunderstanding ...
Will Sawin's user avatar
Will Sawin
  • 149k
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)=\...
Duality's user avatar
Duality
  • 1,541
6 votes
1 answer
544 views

Chebotarev density theorem and pure weight local systems

How do we deduce the following statement from the Chebotarev density theorem? The statement is from Ngo's Fundamental Lemma paper. Let $U$ be a scheme of finite type over $\mathbb{F}_q$. Let $\...
Will Sawin's user avatar
Will Sawin
  • 149k
2 votes
2 answers
469 views

Relation between the Selmer group and the ideal class group

Let $E/K$ be an elliptic curve defined over the number field $K$. Does exist any relation between the $p$-Selmer groups of $E/K$ and the ideal class group $Cl(K)$ of $K$?
Olivier's user avatar
Olivier
  • 10.9k
35 votes
5 answers
9k views

A reference for geometric class field theory?

The classic reference of this topic is Serre's Algebraic Groups and Class Fields. However, many parts of this book use Weil's language, which I find quite hard to follow. Is there another reference ...
user234212323's user avatar
user234212323
  • 894
3 votes
0 answers
187 views

Simplification of links between idele class group and étale cohomology

I posted this question over on stack exchange and was told it would work better here. For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global ...
Unmotivated L-function's user avatar
Unmotivated L-function
  • 31
0 votes
0 answers
140 views

Field of algebraic functions

We assume $K$ as a field of characteristic zero. By a field of algebraic functions of one variable over $K$ we mean a field $R$ satisfying $R=K(x,y)$ with $x$ being transcendental over $K$, and $R$ is ...
Lei's user avatar
Lei
  • 1
4 votes
0 answers
465 views

Reference request: ramified and local geometric class field theory

There are lots of references on global unramified geometric class field theory (following Deligne's $\ell$-adic sheaves approach). There are also some notes talking about how to extend Deligne's ...
wkf's user avatar
wkf
  • 647
5 votes
0 answers
127 views

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 $...
debanjana's user avatar
debanjana
  • 1,283
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 ...
S. carmeli's user avatar
S. carmeli
  • 4,189
5 votes
1 answer
516 views

Reference for Local class field theory via witt vectors

I would like to find some books or lecture notes on geometric local class field theory via Witt vectors. I can't find any good paper on this subject.All approaches in the books to local class field ...
Community's user avatar
CommunityBot
  • 1
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", ...
Community's user avatar
CommunityBot
  • 1
2 votes
0 answers
100 views

Fibers of reciprocity maps and higher dimensional analogs

Part I. Say $K$ is a number field, $v$ is a finite place of $K$, $K_v$ the $v$-adic completion of $K$. We have the local Artin map for every finite $v$: $$\rho_v : K_v^{\times}\to\text{Gal}(K_v^{\...
Community's user avatar
CommunityBot
  • 1
5 votes
1 answer
905 views

Abelianized fundamental group of a curve over a finite field

Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and fix a geometric point $\overline{x} : \text{Spec } \overline{\mathbb{F}_q} \to X$. Then there ...
Community's user avatar
CommunityBot
  • 1
4 votes
0 answers
248 views

Polynomial equations in many variables have solutions (Lang 1952 paper)

I am trying to understand the proof of the following result: Suppose $F$ is a function field in $k$ variables over an algebraically closed field. Let $f_1,...,f_r \in F[x_1,...,x_n]$ be ...
user223794's user avatar
user223794
  • 87
2 votes
0 answers
100 views

Quasi-algebraically closed fields reference request

I am looking for a road to understanding what quasi-algebraically closed fields are with the ultimate goal of understanding the paper by Lang 1952. My current background is the first 6 chapters from ...
user223794's user avatar
user223794
  • 87
5 votes
0 answers
206 views

Real field of definition of an abelian variety of CM-type?

Question 0. Can a field of definitions (without automorphisms) of an (almost arbitrary) abelian variety of CM-type, originally defined over ${\mathbb{C}}$, be chosen to be a totally real number ...
Mikhail Borovoi's user avatar
Mikhail Borovoi
  • 14.2k
2 votes
0 answers
87 views

Rank growth in ray class fields of primes that are inert in an imaginary quadratic extension

If $K=\mathbb{Q}[\sqrt{-d}]$, $E$ is an elliptic curve and $\operatorname{rank}(E(K))=\operatorname{rank}(E(\mathbb{Q}))$, are there infinitely many primes $l$ of $\mathbb{Q}$ so that: (1.) $\...
The Thin Whistler's user avatar
The Thin Whistler
  • 1,805
3 votes
1 answer
326 views

Connected-étale sequence for ordinary CM elliptic curves

Let $E/k$ be an elliptic curve over algebraically closed field of characteristic $p$ with CM, for simplicity, by the maximal order of a quadratic imaginary field $K/\mathbb{Q}$. Suppose that $p$ is ...
Bear's user avatar
Bear
  • 845
12 votes
1 answer
565 views

Parametrizing all cyclic extensions of the rational numbers of degree 5

Is there a polynomial $f(T,X) \in \mathbb{Q}(T)[X]$ in the indeterminate $X$ over the field $\mathbb{Q}(T)$ with $\mathrm{Gal}(f/\mathbb{Q}(T)) \cong \mathbb{Z}/5\mathbb{Z}$ such that for every Galois ...
David Lampert's user avatar
David Lampert
  • 2,665
2 votes
0 answers
102 views

Meromorphic functions on $U^2 = T^3 + 1$, cokernel of $O_S \to F_\infty/O_\infty$ [closed]

See here. Crossposted from math.stackexchange since there's no good answer despite $>$ 20 upvotes. Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(T)$ ...
Community's user avatar
CommunityBot
  • 1
2 votes
0 answers
241 views

Reference request: Cohomology of Elliptic Curves

Is it true that the group $$H^1(Gal(K^{ab}/K)/\mu_{\nu}(Gal(K_{\nu}^{ab}/K_{\nu})),E_{p^n})$$ is always p-divisible? Or are there any conditions which, when satisfied, guarantee its p-divisibility? ...
The Thin Whistler's user avatar
The Thin Whistler
  • 1,805
8 votes
0 answers
220 views

Does the fundamental group identify group structure on subvarieties of products of curves?

Let $C_1,\dots, C_n$ be smooth curves over $\overline{\mathbb F}_p$, not necessarily proper. Let $X$ be a subvariety of $C_1 \times \dots \times C_n$. I'm interested in the natural map: $$ \pi_1^{ab}(...
Will Sawin's user avatar
Will Sawin
  • 149k
0 votes
0 answers
133 views

Reciprocity laws in different dimensions

Let $M/L/Qp$ be a finite galois abelian extension of local fields and define $\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathbb{Z}}a_iT^i:a_i\in M,\min_{i\in \mathbb{Z}}, v(a_i)>−\infty , \lim_{i\to −\...
user56577's user avatar
user56577
  • 33
1 vote
0 answers
101 views

Relation between 1-dimensional and 2-dimensional reciprocity maps

Let $M/L/\mathbb{Q}_p$ be a finite galois abelian extension of local fields and define $\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathcal{Z}}a_iT^i : a_i\in M, \min_{i\in \mathcal{Z}} v(a_i)>-\infty, \...
user56577's user avatar
user56577
  • 33
0 votes
0 answers
100 views

divisor class group with modulus

Let $C$ be a smooth projective curve over a field $k$ and $S \subset C$ a finite number of points. A modulus is simply a divisor supported on $S$. What is the divisor class group with modulus? I ...
sruobera's user avatar
sruobera
  • 1
12 votes
3 answers
2k views

What makes Geometric CFT easier than CFT?

I've been reading: math.stanford.edu/~conrad/249BPage/handouts/geomcft.pdf in an attempt to shed some geometric light on class field theory. The last paragraph there reads: In case the ground field $...
David E Speyer's user avatar
David E Speyer
  • 156k
10 votes
0 answers
758 views

How does one understand geometric CFT in terms of modularity?

I have recently asked a question in a similar vein: What makes Geometric CFT easier than CFT? but I'm afraid I wasn't quite ripe to ask it yet. I have since consulted with the following sources: http:/...
Community's user avatar
CommunityBot
  • 1