Questions tagged [ramification]
The ramification tag has no usage guidance.
57
questions
4
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372 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
0answers
142 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
1answer
49 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 ...
4
votes
0answers
208 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
1answer
184 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
0answers
100 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
1answer
144 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 & \...
4
votes
0answers
190 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
0answers
49 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
1answer
264 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 $...
1
vote
0answers
95 views
Unramified Extensions of Local Fields and $\mathcal{O}_L / Q^e \otimes_{k} k^{sep}$
Suppose I have an extension of number fields $L/K$, and a prime $P$ in $\mathcal{O}_K$ splits into $Q^e$ in $\mathcal{O}_L$. Put $k = \mathcal{O}_K / P$ and $k^{sep}$ for the separable closure of $k$. ...
6
votes
1answer
341 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
0answers
92 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
1answer
816 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 ...
3
votes
1answer
250 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
0answers
132 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
1answer
222 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
1answer
108 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
0answers
208 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
1answer
203 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 ...
0
votes
0answers
118 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
0answers
139 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 $K$. ...
3
votes
0answers
320 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 ...
5
votes
1answer
2k 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 ...
2
votes
1answer
424 views
Ramification of prime ideal in Kummer extension
Let $\mu \in \mathbb{Q}(\zeta_n)$ lie above the rational prime $p$, and let the prime ideal $\mathscr{P}\subset \mathbb{Z}[\zeta_n]$ have ramification index $a$ over $\mu$.
Why is it then true that $...
1
vote
1answer
182 views
Measure of ramification of local fields using upper numbering
We let $F$ be a non-archimedean local field (say with finite residue field). Consider a Galois extensions $E$ of $F$, with $G = Gal(E/F)$, in a fixed separable closure $\bar{F}$ of $F$. Considering ...
2
votes
0answers
689 views
Unramified extensions of a given degree
Let $K \neq \mathbb{Q} $ be a finite extension of $\mathbb{Q}$. For a given integer $n$, how to construct an unramified extension of $K$ of degree $n$ ?
EDIT: If not then under what conditions on $K$,...
5
votes
0answers
184 views
Is the moduli space of curves arising from wild ramification smooth?
Fix a natural number $g$, a prime $p$, and a $p$-group $P$.
Let $C$ be a smooth projective curve of genus $g$ with a faithful action of $P$ and an isomorphism $C / P \cong \mathbb P^1$ such that $P$ ...
10
votes
1answer
401 views
Control ramification in Noether Normalization
Let $X$ be an irreducible affine variety (integral, reduced scheme of finite type over an algebraically closed field $\Bbbk$ of characteristic zero) of dimension $n$. The well-known Noether ...
6
votes
2answers
546 views
When does a dyadic prime ramify in a relative quadratic extension?
In a quadratic extension $\mathbb{Q}(\sqrt{d})$of $\mathbb{Q}$ it is clear that 2 ramifies if and only if $d\equiv 2,3\mod 4$ (easy to see if you compute the discriminant). But if I take a relative ...
2
votes
0answers
174 views
Degeneracy divisor of the “trace” morphism
Let $f\colon X\to Y$ be a finite morphism of smooth curves over an alg. closed field of characteristic zero. I recently asked how methods reminiscent of basic algebraic number theory can be used to ...
9
votes
1answer
392 views
Showing that $2c_1(f_*\mathscr O_X)=-f_*R_f$ on curves, maybe by local fields
I originally asked this question on Mathematic StackExchange, but it did not seem to be attracting any attention, so now I am trying mathoverflow. I hope it is not too simple or unappropriate a ...
2
votes
1answer
301 views
Root discriminant lower bounds in algebraic geometry
Let $X$ be a simply-connected smooth projective variety over $\mathbb C$. Let $C$ be a curve on $X$.
If $Y$ is a ramified cover of $X$ of degree $n$, and $D$ is the branch divisor of $Y$, call $(D \...
6
votes
2answers
903 views
How to explicitly see the ramification over infinity
Take the equation $y^{d}=\Pi_{1}^{n}(x-t_{i})^{m_{i}}$ over $\mathbb{C}$. This affine equation gives a cyclic cover of $\mathbb{P}^{1}$. Now it is usually said without explanation that if the sum $\...
2
votes
3answers
2k views
Basic arithmetic behind ramification in quadratic number fields
Ramification of prime numbers in number fields is a topic relevant to what I'm studying (arithmetic hyperbolic 3-manifolds), and many results from algebraic number theory are used there, however most ...
2
votes
1answer
176 views
ramification of discrete valuation field
Let $K$ be a discrete valuation field with valuation $v:K\rightarrow \mathbb Z\cup \{\infty\}$ which is normalized by $v(\pi)=1$ for a prime element $\pi$. Let $v:\overline K\rightarrow \mathbb Q\cup\{...
2
votes
0answers
174 views
Two different definitions of $\sigma$-L-spaces in Kottwitz I and II
In his papers "Isocrystals with additional structure" I and II, Kottwitz defines the notion of $\sigma$-$L$-spaces. In the first one the situation is the following
$k$ an algebraically closed field ...
5
votes
2answers
1k views
ramified quaternion algebras
I'm trying to better understand the connection between the concepts of ramification of a field extension, and ramification of a quaternion algebra. I'm also trying to build a better understanding of ...
8
votes
1answer
669 views
Question about local description of the branch locus
Let $\pi:Y\to X$ be a dominant, finite morphism of nonsingular varieties over an algebraically closed field $\Bbbk$. Assume furthermore that for all $Q\in Y$, with $P=\pi(Q)$, we have
$$\mathcal O_{Y,...
8
votes
0answers
761 views
Elementary proof of the Hurwitz formula
I am aware of two forms of the Hurwitz formula. The first is more common, and deals only with the degrees. So if $f:X \rightarrow Y$ is a non-constant map of degree $n$ between two projective non-...
7
votes
0answers
177 views
Unicritical rational functions on curves in characteristic $p$
Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$.
How precisely can one describe the ...
6
votes
0answers
476 views
analog for the discriminant of number fields in the function field case?
Is there a nice algebraic way of determining the ramification of a morphism between curves? Ie, some analog of the discriminant of number fields?
Specifically, I'm trying to prove that if $X$ is a ...
4
votes
2answers
911 views
trying to understand the support of the sheaf of relative differentials
So I'm trying to understand a proof of Belyi's theorem from http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf
specifically lemma 3.4.
The setup is as follows: Let $X/\mathbb{C}$ be a curve, and let $t ...
5
votes
2answers
1k views
Fibre cardinality of an unramified morphism
Let $\varphi: X \to Y$ be a finite, dominant, unramified morphism of varieties over an algebraically closed field. If necessary, we can assume $X$ and $Y$ to be nonsingular. I am trying to prove that
...
12
votes
2answers
2k views
Finite, Étale Morphism Of Varieties
I have a, probably very simple, question: My intuition tells me that the following statement should be true, but I couldn't find it anywhere and I wanted to make sure I am not missing something.
Let $...
19
votes
5answers
5k views
Higher dimensional version of the Hurwitz formula?
In Hartshorne IV.2, notions related to ramification and branching are introduced, but only for curves. The main result is the Hurwitz formula.
Now if you have a finite surjective morphism between ...
6
votes
2answers
2k views
Ramification divisor associated to a cover of a regular scheme
Let $S$ be the spectrum of $\mathbf{Z}$ or the spectrum of an algebraically closed field. (Actually, one can take $S$ to be any noetherian integral regular scheme.)
Let $f:X\longrightarrow Y$ be a ...
11
votes
1answer
858 views
Ramification in p-division fields associated to elliptic curves with good ordinary reduction
Dear MO,
Let $p$ be a prime and let $E/\mathbb{Q}_p$ be an elliptic curve. Suppose that $E/\mathbb{Q}_p$ has good ordinary reduction at $p$. In his 1972 paper ``Propriétés galoisiennes des points d'...
2
votes
1answer
478 views
Can we construct rational functions with prescribed ramification on an algebraic curve over \Qbar
Let $C$ be a smooth projective connected curve of genus $g$ over $\bar{\mathbf{Q}}$. Fix a finite non-empty (Edit) set of closed points $S$ in $C$ and let $U$ be the complement of $S$ in $C$.
Q1. (...
8
votes
1answer
616 views
Does combining Abhyankar's Lemma and embedded resolution give horizontal normal crossings
Let $\pi:Y\longrightarrow \mathbf{P}^1_{\mathbf{Z}}$ be a finite surjective flat morphism of schemes, where $Y$ is a normal integral flat projective 2-dimensional $\mathbf{Z}$-scheme, with branch ...
