12
votes

Accepted

### Action of the symmetric group $S_3$ on an elliptic curve $E$ defined over $\mathbb{Z}$

Technically speaking, an elliptic curve is a genus 1 curve with a choice of rational point. The automorphism group of an elliptic curve is the subgroup of automorphisms of the genus 1 curve that fix ...

11
votes

### What are the primes that are ramified?

All the information on the higher ramification groups can be derived from Theorem II.5.6, which TKe notes.
The Galois group of the ray class field is the group of fractional ideals relatively prime ...

10
votes

Accepted

### Cycle type in Galois group from ramified primes

It does not imply there is such a permutation in the Galois group, and your question has an interesting history.
First of all, I would say the question you ask is arguably the wrong one: the more ...

9
votes

Accepted

### What is the effect of imaginary quadratic extension on a quaternion algebra's ramified primes

The base change of your quaternion algebra will be ramified at $\mathfrak{P}$ if and only the degree of the extension of completions $K_{\mathfrak{P}}/F_{\mathfrak{p}}$ has odd degree, i.e. case 3. A ...

8
votes

### Conductor at 2 of abelian surfaces with real multiplication

(Edit: answer below was given before the QM assumption was added)
There are newforms of weight 2 and level 1024 (and trivial nebentypus), e.g., 1024.2.a.a. The associated abelian surface has conductor ...

7
votes

### What are the jumps in the ramification filtration of the absolute Galois group of a local field?

The jumps are all nonnegative rational numbers.
To see that the jumps are only rational numbers, one can use the definition: An integral is involved, but this is an integral of piecewise linear ...

7
votes

Accepted

### Deligne's "Notes sur Euler-Poincaré: brouillon project"

I have a copy of "Notes sur Euler-Poincaré: brouillon project" (which I received from professors Illusie and Saito). I don't feel at liberty to post it on a public web site, but I can email ...

6
votes

Accepted

### Image of the trace map of ring of integers

Question 1. Yes, and this follows from the results of Chapter VIII in Weil: Basic Number Theory. See especially Corollary 2 of Proposition 4 in that chapter.
Question 2. In general, the exponent of $...

6
votes

Accepted

### Is the number of ramified coverings of given degree of a curve with prescribed branch divisor finite?

This can be understood by looking at fundamental groups.
Covers of $Y$ branched only along $B$ of degree $n$ correspond to homomorphisms $\pi_1(Y-B)\rightarrow S_n$ ($S_n$ is the symmetric group on $...

5
votes

Accepted

### Ramification criteria for Kummer extensions

$K( A^{1/n})$ is unramified if and only if the image of $A$ in $K_{\mathfrak p}^* / (K_{\mathfrak p}^*)^n$ lies in a certain cyclic subgroup of order $n$, which is the cyclic subgroup generated by a ...

4
votes

Accepted

### Wild ramification in composite fields

Your intuition is good - it is not true.
We can take $F = \mathbb F_q(t)$ and define $F_1$ and $F_2$ to be adjoining distinct roots of a cubic $x^3 - t x =1$. Then if $E = F_1 ( \sqrt{t})$ since $x, ...

4
votes

### Inverse Galois problem on the upper or lower numbering filtration

It is not sufficient to require that the quotients $G_i/G_{i+1}$ be commutative; they also have to have exponent dividing $p$. There are further restrictions, and the question has been studied by ...

4
votes

Accepted

### Can free rational curves lift to ramified covers of Fano varieties?

After some helpful conversations with Johan de Jong, I came up with an example. After finding it I realize I probably should have figured it out earlier.
In fact, $\operatorname{Hilb}^2(\mathbb P^n)$ ...

4
votes

Accepted

### Shafarevich's conjecture on Galois groups over fields ramified at finitely many places

In the footnote on Page 11 of [Fernando Q. Gouvea: Deformations of Galois Representations],
The reference is
[I.R. Shafarevich, Algebraic number fields, Proceedings of the
international Congress of ...

3
votes

Accepted

### Is the map on tame fundamental groups of a quasi-projective variety, upon base change between algebraically closed fields, an isomorphism?

update: there is now a complete reference [2005.09690] Invariance of the tame fundamental group under base change between algebraically closed fields.
I think it is generally admitted as folklore that ...

3
votes

### A problem in Bushnell and Henniart's book, "The local Langlands conjecture for GL(2)"

If $E/F$ is unramified and $E\neq F$ then the norm map $U_E^m/U_E^{m+1} \rightarrow U^m_F/ U^{m+1}_F$ cannot be an iso, since the groups have different order(= to the order of residue fields.)

3
votes

Accepted

### Swan-conductor and base change

Yes.
We have some finite etale Galois cover $D \to C$ with automorphism group $G$. We can base change this cover, obtaining a finite etale Galois cover $D' \to C'$ with the same automorphism group. ...

3
votes

Accepted

### What is the indecomposable decomposition of holomorphic differentials of an Artin-Schreier curve C as a Z/p-representation?

Thanks to Jeremy Booher for explaining the following in private correspondence. My error was in the indexing in my definition of $v_{1k}$. In their paper, they define
$$v_{1k} := \lfloor \frac{d(P'|P) ...

2
votes

### Deligne's "Notes sur Euler-Poincaré: brouillon project"

That project is mentioned in Illusie's article "From Pierre Deligne’s secret garden: looking back at some of his letters" (Japan. J. Math. 10, 237–248 (2015) DOI: 10.1007/s11537-015-1514-9). ...

2
votes

### Number fields with finite maximal unramified $p$-extensions

One can do this without any recent theorems. Let $\ell$ be a prime congruent to $1$ mod $p$. Then there is a Galois extension $F$ of $\mathbb Q$ with Galois group $\mathbb Z/p$ ramified only at $\ell$,...

2
votes

Accepted

### Characterize the space of all ramification divisors of degree $d$

Here is an elementary answer to your specific question. $R_d$ is not closed. For instance when $X = \mathbb P^1$ and $d = 2$, a divisor $p+q$ appears as a ramification divisor if and only if $p,q$ ...

1
vote

### Characterize the space of all ramification divisors of degree $d$

For non hyper-elliptic curves and $d < g$ you can use a geometric Riemann-Roch argument:
A map $f:C\to\mathbb{P}^1$ of degree $d<g$ is defined by choice of a codimension $2$ linear projective ...

1
vote

Accepted

### Number fields with finite maximal unramified $p$-extensions

The answer to question 2 is also yes. To see this, we need the following two facts.
Fact 1: Let $\zeta_{p^{m}}$ be a primitive $p^m$-th root of unity. Then the class number of $\mathbb{Q}(\zeta_{p^{m}}...

1
vote

### Eisenstein polynomial of totally ramified extension over $p$-adic field

There is a super naive algorithm, assuming that you know some uniformizer $\pi_K$:
Let $\alpha\in L$ be a root of $f$ and $N=\deg(f),M=[K:\Bbb{Q}_p]$. Normalize the valuation such that $v(\pi_K)=N$.
...

1
vote

### The variety induced by an extension of a field

Let $n=2$ and $k=3$, and suppose by the sake of simplicity that the three lines are in general position. Then, up to projective transformations, we can assume that they are the three coordinate lines ...

1
vote

Accepted

### The variety induced by an extension of a field

I decided to turn my comment into an answer not because it is complete but because I think it can be of use.
Let $z=(z_0:z_1:z_2)$ and $u=(u_1:\ldots:u_k)$ be homogeneous coordinates of $\mathbb{P}^2$ ...

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