3
votes
2answers
166 views

Minimal fields of isomorphism for varieties

Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...
4
votes
0answers
241 views

questions about the “relative fundamental group” of SGA 1 Expose XIII

$\newcommand{\LL}{\mathbb{L}}$ I'm reading SGA 1, obtained from http://arxiv.org/abs/math/0206203 My questions regard 4.5 and 4.5.1 (page 309) of Expose XIII. Following "Exemples 4.4" in Expose ...
8
votes
1answer
351 views

Variant of Hilbert 90 for Galois extensions

Let $K/\mathbb F_q(x)$ be a finite Galois extension with Galois group $G$. Let $Aut(K)$ be the group of $\mathbb F_q$-automorphisms of $K$. Obviously, $G\subseteq Aut(K)$. It is well known that ...
1
vote
1answer
121 views

Variety of factorizations of differential operator

Take differential operator as polynomial of letter $d$ with coefficients in some function field, where $d$ act by derivation in this function field. Call it a differential field. For simplicity let ...
2
votes
1answer
195 views

how do automorphisms act on the right in grothendieck's galois theory

So, I'm reading through some notes on the etale fundamental group (mostly Murre, but also some other notes I have), and I find it confusing how in a galois category $\mathcal{C}$ with fundamental ...
6
votes
2answers
410 views

Is there an algorithm to compute efficiently the dessin d'enfant from a Belyi pair?

Let $(X,f)$ be a Belyi pair, i.e. a Riemann surface $X$ together with a morphism $f: X \to \mathbb{P}^1$, ramified only in $0,1, \infty$. Grothendieck's dessin d'enfant is the pre-image $G$ of the ...
12
votes
1answer
630 views

Janelidze's Galois theory

I am interested in learning about categorical Galois theory, as developed by Janelidze. I am a graduate student who has good familiarity with category theory, but not in the level of doing research on ...
2
votes
2answers
200 views

Confusion about two statements about cohomology of curves with automorphisms

Let $\pi :C\rightarrow \mathbb{P}^{1}$ be a cyclic cover of degree $m$ of $\mathbb{P}^{1}$. So $C$ has an action of $\mathbb{Z}/m\mathbb{Z}$. Let $\xi$ be a primitive $m$-th root of unity. Consider ...
2
votes
0answers
52 views

lift isomorphic in a sufficiently thick fiber

Let $P$ and $P'$ two polynomials in $k[[\pi]][t]$ for an algebraically closed field $k$ and let $A=k[[\pi]]$. We consider $ X'=Spec (A[t]/(P'))$ and $X=Spec (A[t]/(P)$. Let $d=val(\Delta(P))$ where ...
3
votes
1answer
276 views

Rigidity, moduli space, and moduli field

In his comment to the question Algebraic numbers and the complex projective line minus three points JSE says that for algebro-geometric objects defined over the complex numbers "in practice, 'the ...
2
votes
0answers
98 views

On Artin-Hironaka lemma and Galois theory

Let $A=k[[t]]$ Let $B$ a flat $A$-finite algebra which is etale and Galois at the generic point. Then by Artin lemma 3.12 (ii) in his IHES paper on approximation, we know that there exists an integer ...
0
votes
1answer
173 views

Hurwitz's construction of simple covers

What is commonly meant by Hurwitz's construction of simple covers?
3
votes
0answers
148 views

Correspondences of schemes induced by a finite Galois extension

I am trying to do the first exercises of the Mazza-Voevodsky-Weibel book Lecture Notes on Motivic Cohomology, which are some elementary results about correspondences between schemes. Recall that if ...
21
votes
2answers
731 views

Profinite groups as étale fundamental groups

Does every profinite group arise as the ├ętale fundamental group of a connected scheme? Equivalently, does every Galois category arise as the category of finite ├ętale covers of a connected scheme? ...
1
vote
4answers
674 views

Explicit element in free group which is killed by every solvable quotient

The free group on two generators $F_2=\langle x,y|\rangle$ is the fundamental group of $\mathbb P^1(\mathbb C)\setminus\{0,1,\infty\}$. Now, there are plenty of galois covers of this space whose ...
8
votes
1answer
900 views

Is every finite group a quotient of the Grothendieck-Teichmuller group?

The Grothendieck-Teichmuller conjecture asserts that the absolute Galois group $Gal(\mathbb{Q})$ is isomorphic to the Grothendieck-Teichmuller group. I was wondering, would this conjecture imply the ...
1
vote
1answer
201 views

Is the other extreme of Hilbert Irreducibility true?

Let $K$ be a number field (or perhaps more generally a Hilbertian field). Let $X_K\rightarrow \mathbb{P}^1_K$ be a regular (i.e. without extension of scalars) $G$-Galois branched cover. Hilbert's ...
12
votes
2answers
2k views

Why should the anabelian geometry conjectures be true?

I had probed friends of mine about Grothendieck's motivation for making the anabelian geometry conjectures, and they gave me the following explanation: If $X$ is a hyperbolic curve over some field ...
6
votes
1answer
318 views

Galois groups at closed points from Galois group at generic point?

Consider the finite map $\mathbb{A}^1_\mathbb{Q}\rightarrow \mathbb{A}^1_\mathbb{Q}$ given by $z\mapsto z^5-z$. The fiber over generic point is the field extension $\mathbb{Q}(t)[z]/(z^5-z-t)$ over ...
3
votes
1answer
473 views

What is the general statement of Hilbert 90?

I know two generalizations of Hilbert 90, but I don't if there is a statement that contains both: The first statement Let $K$ be a field, then $H^1(Gal(K), GL_n(K^{sep}))=0$. The second statement ...
7
votes
2answers
935 views

What are the different theories that the motivic fundamental group attempts to unify?

I must preface by confessing complete ignorance in the subject. I've read introductory texts about the theory of motives, but I am certainly no expert. In ...
8
votes
3answers
667 views

What is the purpose of tangential base-points?

Let $V$ be an affine complex variety. Let $x \in V$ be a closed point. Then a tangential base-point at $x$ is $x$ together with a regular function $t$ on $V$ that is zero exactly on $x$ (to degree ...
5
votes
2answers
403 views

Expressing Galois actions on fundamental groups explicitly

Let $X$ be some variety over $\mathbb{Q}$, and let $\pi_1(X\times_{\mathbb{Q}}\mathbb{C},x)$ denote its (topological) fundamental group. As is well known $Gal(\mathbb{Q})$ acts on this fundamental ...
2
votes
1answer
153 views

In Riemann Existence, what is the interpretation of the space of complex-geometric points?

I've been thinking recently about moduli spaces defined over $\mathbb{Z}$, and this led me to the following question: Question Riemann existence says that if we have a variety over $\mathbb{C}$, ...
22
votes
5answers
2k views

Grothendieck's Galois theory without finiteness hypotheses

This is motivated by the discussion here. The usual definition of the etale fundamental group (as in SGA 1) gives automatic profiniteness: Grothendieck's formulation of Galois theory states that any ...
46
votes
0answers
3k views

Grothendieck-Teichmuller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures Grothendieck-Teichmuller conjecture: the morphism $$ G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T}) $$ is an isomorphism. Here ...
8
votes
2answers
987 views

Where is a good place to start learning about the Grothendieck-Teichmuller group?

I've had a desire to get some sort of handle on the Grothendeick-Teichmuller group for years now, but I've always felt that I could never find a source that was introductory and readable. The obvious ...
5
votes
1answer
372 views

Given a branched cover with branch cycle description $(g_1,…,g_r)$, does $g_i$ generate some decomposition group?

Classically: Let $a_1,...,a_r$ be points in $\mathbb{P}^1_{\mathbb{C}}$, and let $\alpha_1,...,\alpha_r$ be simple loops around the $a_i$, all counterclockwise, and none touching (so ...
1
vote
1answer
156 views

Behaviour of Primes under Regular Coefficient Extensions

Let $K\hookrightarrow K'$ be a regular extension of fields, and $K[x_{1},\cdots,x_{n}]\hookrightarrow K'[x_{1},\cdots,x_{n}]$ the corresponding ring extension. Does every prime ideal of the first ring ...
6
votes
3answers
614 views

Branch locus of the Galois closure of a Belyi morphism

A morphism of curves is said to be Galois if the corresponding extension of function fields is Galois. Let $f:X\longrightarrow Y$ be a finite morphism of smooth projective connected curves over ...
1
vote
1answer
210 views

Image of spliting of short exact sequence of algebraic fundamental groups

If we have a variety X , over a field k, and x is a geometric point of X, and let $\bar x $be a geometric point of $X_{k^s} := X \times_k k^s $above x then we have the following short exact sequence: ...
5
votes
1answer
541 views

Is it possible to recover the degree of a field extension from a list of elements and the ground field?

I'm interested to know if there is anything known about recovering the degree of a field extension, $E/k$, given $E=k(\alpha_1,\ldots, \alpha_n)$ (here I'm assuming that the extension is of finite ...
11
votes
2answers
443 views

Realizing D_8 as a Galois group over C(x) with prescribed decomposition groups

Coming up with examples of $D_8$-covers of $\mathbb{C}(x)$ is easy. For example: $Quot(\mathbb{C}(x)[y,z]/(y^2=x(x-7), z^4=(y+\sqrt{-6})^2(y-\sqrt{-6})^2(y+\sqrt{-10})(-y+\sqrt{-10})^3))$ defines a ...
2
votes
1answer
666 views

How do I visualize finite covers of curves over non-algebraically closed fields?

If $L$ is algebraically closed, fields of transcendence degree one over $L$ correspond to algebraic curves over $L$ up to birational equivalence, and finite extensions correspond to finite Galois ...
10
votes
5answers
3k views

Grothendieck's Galois Theory today

I have recently become aware of, and started to study in my free time (abundant in these summer months) Grothendieck's Galois Theory (GGT), as formulated in SGA 1 and later by Grothendieck's ...
7
votes
2answers
555 views

A split short exact sequence of algebraic fundamental groups

If we have a variety, $X$, over a field, $k$, and $x$ is a geometric point of $X$, and let $\bar x$ be a geometric point of $X_{k^s} := X \times_k k^s$ above $x$ then we have the following short exact ...
11
votes
0answers
594 views

Does there exist a number field, unramified over a predetermined finite set of primes of Q, such that the inverse regular Galois problem is correct for that number field?

The question is: for any finite group, $G$, and any finite set of primes (of $\mathbb{Z}$), $P$, is there a number field $K$, such that there is a regular $G$-Galois extension of $\mathbb{P}^1_K$, and ...
18
votes
3answers
2k views

Surprising Analogue of Q

I was describing Manish Kumar's work a few weeks ago to a fellow graduate student, and she stumped me with a big-picture question I couldn't answer. Manish Kumar proved that the commutator subgroup ...
6
votes
1answer
299 views

Computing stable reduction of finite covers of curves in practice

The general theory is described in various places, but I'll be following (sketchily) the description of this process appearing in section 1 of Bouw and Wewers' "Reduction of covers and Hurwitz ...
1
vote
2answers
288 views

sheaves of representations on galois groups, can there be interesting cohomology?

Consider a field $K$ (of characteristic 0, say) and its absolute galois group $G_K^{ab} = Gal(\overline{K}/K)$, given the Krull topology: $U_E(\sigma) = \sigma Gal(\overline{K}/E)$ form a basis of the ...
6
votes
1answer
221 views

For a given finite group G, is there a cover of P^1 over Q s.t. over C it's G-Galois?

For any finite group, G, we can find a cover of ℙ1ℂ which is G-Galois. The regular inverse Galois problem is equivalent to there existing such a cover that descends with action to ℚ. ...
48
votes
2answers
3k views

The inverse Galois problem and the Monster

I have a slight interest in both the inverse Galois problem and in the Monster group. I learned some time ago that all of the sporadic simple groups, with the exception of the Mathieu group $M_{23}$, ...
8
votes
2answers
794 views

What, precisely, is the relationship between “fields of moduli” and “moduli spaces”?

Notation The term "field of moduli" comes in up in different scenarios, but let's consider the following: Let X->ℙ1 be a G-Galois cover, where everything is over the algebraic closure of some ...
4
votes
1answer
294 views

Field of Definition of a Meromorphic Function

Question Let X be a smooth, projective curve over the algebraic closure of ℚ. Let f:X->ℙ1 be a meromorphic function. Assume that the zeros and the poles are defined over some number ...
16
votes
1answer
953 views

A Galois Theory Computation

Excuse me for the specificity of this question, but this is a silly computation that's been giving me trouble for some time. I want to explicitly realize the order 21 Frobenius group over ℂ(x), ...