# Questions tagged [ramification]

The ramification tag has no usage guidance.

83
questions

8
votes

1
answer

302
views

### Regarding upper numbering of ramification groups

In Serre's book "Local fields" he defines the function $\phi(u)=\int_{0}^{u}\frac{dt}{( G_0:G_t)}$ and defines the upper number of ramification groups as $G^v=G_{\phi^{-1}(v)}$ and somehow ...

4
votes

0
answers

115
views

### Introduction to the theory of $D$-modules and the role of the characteristic cycle

I am seeking recommendations for a concise introduction to the theory of $D$-modules suitable for an algebraic geometer. Specifically, I am interested in understanding:
The role of the characteristic ...

2
votes

0
answers

156
views

### Unramified lisse $\overline{\mathbb{Q}}_{\ell}$-sheaves

Let $X$ be a connected noetherian scheme and $\ell$ a prime invertible on $X$. Let $D \subset X$ be a regular effective Cartier divisor (or more generally a normal crossings divisor, I suppose). Write ...

3
votes

1
answer

261
views

### Ramification criteria for Kummer extensions

Let $K$ be a number field containing $n$-th roots of unity. The usual Kummer theory provides a correspondence between between abelian subgroups $A \subset K^*/(K^*)^n$ and abelian extensions of K of ...

5
votes

2
answers

241
views

### Characterize the space of all ramification divisors of degree $d$

Let $X$ be a compact Riemann surface of genus $g>0$, and let $f\colon X \to \mathbb{P}^1$ be a branched covering of degree $d$. Define the ramification divisor $R_f$ on $X$ by $f$, where $\deg R_f =...

1
vote

1
answer

194
views

### Is the map on tame fundamental groups of a quasi-projective variety, upon base change between algebraically closed fields, an isomorphism?

$\DeclareMathOperator\Spec{Spec}
$Let $k \subset L$ be two algebraically closed fields of characteristic $p$. Let $U \subset \mathbb P^1_k$ be a smooth quasi-projective curve and let $U_L$ denote the ...

0
votes

0
answers

50
views

### The decomposition forms of primes in $A_5$-fields

Let $K$ be a number field of degree $5$ whose Galois closure (over $\mathbb{Q}$) has the Galois group $A_5$, the alternating group of degree five. Is there any result concerning the decomposition ...

1
vote

0
answers

85
views

### Inflation-restrction sequence for maximal $S$-ramified extension

Let $K$ be a number field. Let $G_K$ be an absolute Galois group of $K$. Let $M$ be a $G_K$-module and $L/K$ be a finite extension.
There is a inflation-restriction exact sequence,
$0\to H^1(Gak(L/K), ...

5
votes

1
answer

309
views

### Conductor at 2 of abelian surfaces with real multiplication

Let $A/\mathbb{Q}$ be an abelian surface such that $\text{End}_{\mathbb{Q}}(A)\otimes \mathbb{Q}$ is a real quadratic field $E$. I am interested in bounding the conductor of $A$ at $2$.
Let $\mathfrak{...

6
votes

0
answers

136
views

### Finiteness of wildly ramified cohomology

$\newcommand\p[1]{\left(#1\right)}\newcommand\Char{\operatorname{char}}\newcommand\Gal{\operatorname{Gal}}\newcommand\b[1]{\left\{#1\right\}}$
Let $K$ be a global field. All cohomology below is fppf-...

5
votes

2
answers

523
views

### Number fields with finite maximal unramified $p$-extensions

As discussed in the question number fields with no unramified extensions, it is an open question whether there are an infinite number of number fields that have no unramified extensions. Inspired by ...

4
votes

1
answer

246
views

### Shafarevich's conjecture on Galois groups over fields ramified at finitely many places

Let $K$ be a number field i.e. a finite extension of $\mathbb{Q}$, $\overline{K}$ a fixed separable closure of $K$, and $G_K:=\mathrm{Gal}(\overline{K}/K)$ the absolute Galois group of $K$. Let $S$ be ...

4
votes

1
answer

386
views

### What are the jumps in the ramification filtration of the absolute Galois group of a local field?

Let $k$ be a (complete) discretely valued field and $\ell$ a Galois extension of $k$, possibly infinite. The Galois group $\Gamma=\text{Gal}(\ell/k)$ of $\ell$ over $k$ admits a descreasing, $\mathbb ...

5
votes

1
answer

481
views

### Cycle type in Galois group from ramified primes

Let $P \in \mathbb Z[X]$ be monic, separable, of degree $d$, $K$ its splitting field over $\mathbb Q$ and $G$ the Galois group of $K$ over $\mathbb Q$.
Now, let $p$ be a prime number unramified in $K$....

2
votes

0
answers

88
views

### What is the relationship between ramification in central simple algebras and in fields?

Suppose $K$ is the field of fractions of a Dedekind domain $R$, and let $L$ be a finite extension of $K$. There is a notion of ramification of primes of $K$ in $L$, which describes how $\mathfrak p \...

5
votes

0
answers

170
views

### defining the upper ramification numbering

Given a local field $K$ with absolute Galois group $\Gamma$. Is it "possible" to define the upper numbering on $\Gamma$ without using the lower numbering?
In other words, given $\gamma \in \...

3
votes

0
answers

138
views

### What direction does the derivation of an inseparable algebraic variable point in?

I've been thinking about the geometry of inseparable field extensions lately, since I'm studying smoothness in commutative rings in an advanced topics course this semester. I've generally come to the ...

2
votes

1
answer

469
views

### Eisenstein polynomial of totally ramified extension over $p$-adic field

Let $p\geq 3$ be a prime number, $K$ be a finite extension of $\mathbb{Q}_p$ with no non-trivial unramified subextension, $f(x)$ be an irreducible monic polynomial in $\mathcal{O}_K[x]$, making $L=K[x]...

3
votes

0
answers

183
views

### Wildly ramified extension field

Given an algebraically closed complete valued field $(k,|.|)$ with characteristic 0, such that the residue field $\tilde{k}$ has a positive characteristic, and consider the complete extension $(\...

1
vote

0
answers

190
views

### Is the following local map unramified?

Let $(R,m)$ and $(S,n)$ be two local rings, $R \subseteq S$, $R$ regular, $S$ a finitely generated and flat $R$-algebra, and $mS=n$.
In comments to this question it was claimed that in such situation ...

2
votes

0
answers

104
views

### Weierstrass points in Artin-Schreier extensions of the rational function field in characteristic $p$

I encountered a problem when proving a result regarding the orders at non-Weierstrass points ($n$ is an order of the point $P$ if there is some holomorphic differential $\omega$ with order $n$ at $P$) ...

9
votes

1
answer

485
views

### What is the indecomposable decomposition of holomorphic differentials of an Artin-Schreier curve C as a Z/p-representation?

I am attempting to decompose the holomorphic differentials of an Artin-Schreier-Witt curve as a $\mathbb{Z}/p^n$-representation. This is done in Theorem 1 of Madan-Valentini Automorphisms and ...

2
votes

1
answer

199
views

### Wild ramification in composite fields

Let $ F $, $ F_i $ for $ i = 1, 2 $ and $ E $ be function fields over a finite field with characteristic $ p $ such that $ F \subseteq F_i \subseteq E $ and $ E = F_1 \cdot F_2 $ is the composite ...

5
votes

1
answer

487
views

### Action of the symmetric group $S_3$ on an elliptic curve $E$ defined over $\mathbb{Z}$

I came up with the following question on a facebook group: find the positive integer solutions of the equation $$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$
Now clearly this is very difficult, ...

1
vote

0
answers

79
views

### A question about Hitchin discriminant

Let $X$ be a smooth projective curve over $\mathbb{C}$, $\mathcal{M}$ be moduli space of Higgs bundle of rank $r\geq2$ and degree $d$ on $X$, $W=\bigoplus_{i=2}^{r}H^{0}(X,K_{X}^{\otimes i})$, and $H\...

3
votes

0
answers

187
views

### branch divisor of this map

We consider the blow up $Bl(\mathbb{P}^2)_p$ of $\mathbb{P}^2$ in $p:=|1:0:0|$ and the following surface:
$Y:=\{(|y_1: y_2:y_3:y_4|, |x_0:x_1:x_2|) \in \mathbb{P}^3\times \mathbb{P}^2: rk(\begin{...

6
votes

2
answers

695
views

### The variety induced by an extension of a field

If $K$ is a finitely generated field extension of $k$, then there exists an irreducible affine $k$-variety with function field $K$. The idea is that if $x_1, \dots, x_n$ are generators of $K$ under $k$...

4
votes

0
answers

403
views

### Is there a Galois theory for deformations of curves?

I have some general questions about the deformations of Galois covers of curves. Suppose we are given a $G$-Galois cover $k[[z]]/k[[x]]$, where $k$ is algebraically closed of characteristic $p>0$. ...

1
vote

0
answers

293
views

### Computing the kernel of some Artin-Map

let $K = \mathbb{Q}(\sqrt{-5})$ and $L = K(i)$. $\mathcal{O}_K$ is the ring of integers of K.
I would like to show that the kernel of the Artin-Map $\phi_{L/K}: I_K \rightarrow Gal(L/K)$ is $P_K$, ...

-1
votes

1
answer

114
views

### Relative degree of a prime over a number field (notation from Algebraic Number Fields from Gerald J. Janusz) [closed]

I´m working with "Algebraic Number Fields" from Gerald J. Janusz (1. edition from 1973) and I have a question about his notation.
In chapter IV proposition 4.5 he states if K is an algebraic number ...

11
votes

2
answers

714
views

### Deligne's "Notes sur Euler-Poincaré: brouillon project"

Is the note "Notes sur Euler-Poincaré: brouillon project" by Deligne available somewhere online? If not, is there a way that I could get a copy of it?

2
votes

1
answer

281
views

### A problem in Bushnell and Henniart's book, "The local Langlands conjecture for GL(2)"

On page 123 of Chapter 5 in Bushnell and Henniart's book The Local Langlands Conjecture for GL(2), they state
an elementary property of tamely ramified extension of local fields, which is as follows,
...

3
votes

0
answers

143
views

### Values of Grössencharacter attached to CM elliptic curve

I am trying a cross-post here, as my previous post on stackexchange was not as fruitful as I hoped. The link to the older post is: https://math.stackexchange.com/questions/3327269/values-of-...

2
votes

1
answer

263
views

### Ramification divisor with base change

Let's work over $\mathbb{C}$. Consider the following commutative diagram
\begin{array}{llllllllllll}
E_1& \xrightarrow{f} &E_2\\
\downarrow{\pi} &&\downarrow{\pi}\\
P_1 & \...

3
votes

0
answers

540
views

### Ramification concept in complex analysis and algebraic number theory

I have a question about the connection between the concept of ramification/branching out for Riemann surfaces and algebraic number theory:
In the theory of Riemann surfaces we have following ...

1
vote

0
answers

57
views

### Infinite Non Abelian Extensions Unramified Outside p

Let $K$ be a number field and $p$ be a fixed odd prime. Suppose $\mathfrak{p}\mid p$ is the only prime prime above $p$ in $K$, and that $p$ does not divide the class number of $K$ (I am okay with ...

1
vote

1
answer

608
views

### Swan-conductor and base change

Let $C$ be a proper smooth curve over a perfect field $K$ of positive characteristic $p$, $u: U \hookrightarrow C$ strictly open and $\mathfrak{F}$ a lisse (lcc) $\mathbb{F}_l $-sheaf $(l \neq p)$ on $...

6
votes

1
answer

535
views

### Can free rational curves lift to ramified covers of Fano varieties?

Does there exist $X$ a smooth Fano manifold, $f: Y \to X$ a nontrivial ramified finite cover, $C \subseteq X$ a smooth very free rational curve, such that $f$ is étale over a neighborhood of $C$?
...

3
votes

0
answers

202
views

### Formally unramified morphisms and open diagonals

Let $R\to S$ be a commutative unital $R$-algebra with dual arrow $X\to Y$. If I understand correctly, an open diagonal $\Delta_{X/Y}:X\to X\times _YX$ always implies $X\to Y$ is formally unramified, ...

10
votes

1
answer

1k
views

### What are the primes that are ramified?

Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. We know that the maximal unramified extension (Hilbert class field) $H/K$ is $K(j(E))$. Can we ...

5
votes

1
answer

514
views

### Image of the trace map of ring of integers

Let $L/\mathbb{Q}$ be a finite Galois extension, and let $\mathcal{O}_L$ be the ring of integers of $L$.
We have $tr_{L/\mathbb{Q}}(\mathcal{O}_L)=d\mathbb{Z}$ for some $d\geq 1.$
Fact. $d=1$ if ...

1
vote

0
answers

148
views

### Reference request about what ramification looks like etale locally

Specifically, I'm looking for the proof of the following statement:
If $X\rightarrow Y$ is a finite map of degree $n$ of curves over $\mathbb{C}$, and $P\in X$ maps to $Q\in Y$ with degree of ...

2
votes

1
answer

415
views

### Is the number of ramified coverings of given degree of a curve with prescribed branch divisor finite?

Let Y be a smooth projective curve over C and prescribe a branch divisor B on Y. I want to know if the number of coverings of Y of fixed degree and branched along B is finite. If so, why? Or, where is ...

4
votes

1
answer

137
views

### Inverse Galois problem on the upper or lower numbering filtration

Let $K$ be a (complete) discrete valuated field and $E$ a Galois extension of group $G$. Then one has two filtrations on $G$, the upper and the lower numbering.
Assume that $K$ is equal to its ...

0
votes

0
answers

605
views

### Ramified covering interpretation of an elliptic curve

Let $E:y^2=x(x-1)(x-\lambda)$ be the Legendre form of an elliptic curve $E$ defined over $\mathbb{C}$. The ramified covering $E\to \mathbb{P}_{1}$ defined so that $(x,y)\mapsto x$ has two branches and ...

2
votes

1
answer

302
views

### What is the effect of imaginary quadratic extension on a quaternion algebra's ramified primes

A smart man once explained to me how to solve the following problem, then I forgot.
Let $F\subset\mathbb{R}$
be a number field,
let $d\in F^+$,
and let $K=F(\sqrt{-d})$.
Denote the rings of integers ...

1
vote

0
answers

149
views

### Existence of infinite or bad places that ramify in $K(p^{-1}E(K))/K$ where $p$ is a prime of good reduction

Let $E$ be an elliptic curve, $K$ a number field so that $\operatorname{rank}_{K}(E)\geq 1,$ $p$ a prime of $\mathbb{Q}$ at which $E$ reduces well.
We know (see for instance Silverman, The Arithmetic ...

5
votes

0
answers

162
views

### Traces in finite extensions of integrally closed domains

$\def\fp{\mathfrak{p}}\def\fq{\mathfrak{q}}$I'm looking for a reference for the following commutative algebra fact.
Let $A$ be an integrally closed integral domain, with field of fractions $K$. Let $...

3
votes

0
answers

421
views

### Relation between ramification index and length of filtration of ramification groups

Given a complete valued field $K$ with a discrete value group $\mathbb{Z}$, consider a totally ramified finite Galois extension $L$ of $K$ with its Galois group $G$. Let $O_L$ be the valuation integer ...

7
votes

1
answer

4k
views

### Unramified extension of number fields

Any finite field extension (in particular Galois extension) of $\mathbb{Q}$ is ramified. Is there an intuitive geometric explanation of this fact?
Suppose we have an number field $K$, is any Galois ...