# 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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### 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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### 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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### 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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### 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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### 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
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### 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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### 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
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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 ...
1 vote
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### 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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### 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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### 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 ...
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,...