# Questions tagged [ramification]

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $v:\overline K\rightarrow \mathbb Q\cup\{... 0answers 174 views ### Two different definitions of$\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$an algebraically closed field ... 2answers 1k views ### ramified quaternion algebras I'm trying to better understand the connection between the concepts of ramification of a field extension, and ramification of a quaternion algebra. I'm also trying to build a better understanding of ... 1answer 669 views ### Question about local description of the branch locus Let$\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,... 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-... 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 ... 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 ... 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 ...
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 ...