Questions tagged [ramification]
The ramification tag has no usage guidance.
83
questions
8
votes
1
answer
295
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
113
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 ...
- 2,811
2
votes
0
answers
152
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 ...
- 53
3
votes
1
answer
248
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 ...
- 5,780
5
votes
2
answers
240
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 =...
- 713
1
vote
1
answer
189
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 ...
- 13
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 ...
- 385
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), ...
- 1,467
5
votes
1
answer
308
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{...
- 889
6
votes
0
answers
133
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-...
- 161
5
votes
2
answers
520
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 ...
- 561
4
votes
1
answer
245
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
375
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 ...
- 509
5
votes
1
answer
477
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 $K$....
- 1,303
2
votes
0
answers
86
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 \...
- 121