Questions tagged [ramification]

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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 ...
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2 votes
1 answer
217 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]...
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3 votes
0 answers
105 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 $(\...
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1 vote
0 answers
164 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 ...
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2 votes
0 answers
78 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 $P$) ...
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9 votes
1 answer
420 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 ...
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2 votes
1 answer
112 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 ...
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5 votes
1 answer
355 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, ...
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1 vote
0 answers
72 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\...
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0 answers
123 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{...
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6 votes
2 answers
398 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 $k$...
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4 votes
0 answers
393 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 $p>0$. ...
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1 vote
0 answers
198 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 $P_K$, ...
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-1 votes
1 answer
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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 ...
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4 votes
0 answers
248 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?
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2 votes
1 answer
230 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, ...
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3 votes
0 answers
115 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-...
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  • 81
2 votes
1 answer
202 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 & \...
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3 votes
0 answers
385 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 ...
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1 vote
0 answers
52 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 ...
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  • 1,059
1 vote
1 answer
385 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 $...
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  • 63
1 vote
0 answers
135 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 $k$. ...
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6 votes
1 answer
439 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 $C$? ...
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3 votes
0 answers
151 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, ...
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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 ...
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  • 1,059
4 votes
1 answer
344 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 ...
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  • 1,349
1 vote
0 answers
136 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 ...
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2 votes
1 answer
309 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 ...
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4 votes
1 answer
112 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 ...
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  • 423
0 votes
0 answers
354 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 ...
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  • 615
2 votes
1 answer
250 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 ...
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  • 2,368
1 vote
0 answers
141 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 ...
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5 votes
0 answers
152 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 $...
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3 votes
0 answers
379 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 ...
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7 votes
1 answer
3k 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 ...
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  • 1,543
2 votes
1 answer
569 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 $...
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  • 313
1 vote
1 answer
229 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 ...
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  • 303
2 votes
0 answers
813 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 $K$,...
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  • 1,171
5 votes
0 answers
188 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 $P$ ...
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  • 115k
10 votes
1 answer
486 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 ...
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6 votes
2 answers
606 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 ...
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2 votes
0 answers
186 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 ...
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9 votes
1 answer
419 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 ...
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2 votes
1 answer
354 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 \...
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  • 115k
7 votes
2 answers
1k 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 $\...
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  • 2,131
2 votes
3 answers
3k 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 ...
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  • 2,368
2 votes
1 answer
187 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\{...
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  • 93
2 votes
0 answers
187 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 ...
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  • 271
5 votes
2 answers
2k 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 ...
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  • 2,368
8 votes
1 answer
711 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,...
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