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11
votes
3answers
449 views

Is a cover Galois if and only if it is geometrically Galois

Let $K$ be a number field and let $\pi:X\to \mathbf{P}^1_K$ be a finite morphism, where $X$ is a smooth projective geometrically connected curve. Is $\pi$ a Galois cover if and only if the base ...
0
votes
0answers
140 views

What is the Hurwitz number of an elliptic curve

One can associate a Hurwitz number to any rational function $f:X\to \mathbf{P}^1$ on a compact connected Riemann surface $X$ which ramifies over precisely FOUR points. Suppose that $X$ is an elliptic ...
0
votes
0answers
74 views

What is the moduli space of smooth AFFINE curves of given genus

It is interesting to study the moduli space $M_g$ of smooth projective curves of genus $g$. Why not smooth affine curves of genus $g$?
3
votes
0answers
78 views

Does composing the Frobenius with an automorphism give another Frobenius

Let $X_0$ be a variety over $\mathbf{F}_q$. Consider the Frobenius $F_0:X_0\to X_0$. Let $X= X_0\times \bar{\mathbf{F}_q}$ and let $F:X\to X$ be $F_0 \times \textrm{id}$. Let $f:X\to X$ be an ...
3
votes
0answers
112 views

Why is the trace map on an abelian variety continuous

Let $X$ be a (EDIT) variety with a group structure. For $a\in A$, let $t_a$ be the translation on $X$: $t_a(x) = a+x$. Why is the function $f:X\to \mathbf{C}$ given by $$f(a) = \sum_{i} (-1)^i ...
2
votes
1answer
139 views

how does one intersect the diagonal with a graph on the surface $X\times X$

I want to do a concrete example of an intersection product for myself. Consider the endomorphism $f:\mathbf{P}^1_k\to \mathbf{P}^1_k$ given by $(x:y)\to (y:x)$. It has precisely two fixed points: ...
5
votes
1answer
238 views

The normalization of the peculiar curve $x^p + y^p - (x+y)^p = 0$

Fix a prime number $p$. Consider the affine curve $C$ in $\mathbf{A}^2$ over a number field $K$ given by the equation $x^p+y^p - (x+y)^p =0$. Its Jacobi matrix is $(px^{p-1} -p(x+y)^{p-1} \ py^{p-1} ...
3
votes
0answers
168 views

Minimal resolution of singularities of Fermat curve

Fix a prime number $p$. Let $F$ be the geometrically connected Fermat curve of exponent $p$ over a number field $K$, i.e., $F$ is given by the equation $x^p+y^p = z^p$. Consider the same equation ...
3
votes
1answer
169 views

units in discrete valuation rings

Let $B/A$ be an extension of discrete valuation rings such that the purely inseparable extension of residue fields $l/k$ has a primitive element, i.e., $l=k(y)$ for some element $y$ in $l$. I want to ...
6
votes
1answer
154 views

Computing the trace of the following automorphism of the elliptic curve $y^2 = x^3+x$

Consider the elliptic curve $E$ defined by $y^2z= x^3 +xz^2$ over an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$. Consider the endomorphism $f:E\to E$ given by $(x:y:z)\mapsto ...
12
votes
1answer
357 views

Working with Morphisms in Local Coordinates

In light of the holiday, I would like to air a grievance. I have no good way to recoordinatize a morphism of varieties as I move between coordinate neighborhoods. Let me explain what I mean with ...
13
votes
1answer
340 views

2-Torsion Group Scheme

Consider the elliptic curve $zy^2 + z^2y = x^3.$ I would like to explictly compute the 2-torsion group scheme, $E[2],$ over $\mathbf{Spec}(\mathbb{Z}_2),$ but I'm having a tough time writing down the ...
1
vote
2answers
128 views

can singular points become nonsingular after a base change

Let $X$ be a normal surface over a field $k$. Assume that $X$ is singular. Does there exist a field extension $L/k$ (finite or infinite) such that $X_L$ is nonsingular? The answer is no in general. ...
10
votes
2answers
412 views

In what senses are archimedean places infinite?

According to Bjorn Poonen's notes here (§2.6), we should add the archimedean places of a number field $K$ to $\operatorname{Spec} \mathscr{O}_K$ in order to get a good analogy with smooth projective ...
3
votes
1answer
186 views

Characterization of Horizontal Irreducible Divisors

I´m asking for a proof of a fact used by Arakelov in his paper: Intersection Theory of Divisors on an Arithmetic Surface (page 1169 row 16). He gives no references or explanations for this fact. The ...
10
votes
1answer
132 views

Which number fields allow higher genus curves with everywhere good reduction

The field of rational numbers is not such a number field. That is, there does not exist a smooth projective morphism $X\to\text{Spec } \mathbf{Z}$ such that the generic fibre is a curve of genus ...
4
votes
1answer
197 views

discriminant of an étale cover of an elliptic curve

Let $\pi:X\to E$ be a finite étale morphism, where $E$ is an elliptic curve over a number field $K$. Assume $X$ to be connected, and to be of genus 1. Edit: Assume $X$ and $E$ have semi-stable ...
7
votes
1answer
169 views

Twists of curves over number fields

Let $X$ be a curve over $\overline{\mathbf{Q}}$. I can prove that $X$ can be defined over some number field. (Take two equations defining $X$ in $\mathbf{P}^3$ and consider the number field ...
0
votes
0answers
170 views

a question on the Poincaré bundle

Let $C$ be a smooth curve. Letting $J$ be its Jacobian, consider the Poincaré bundle $\mathcal P$ on $J\times J$. Let $p: J\times J\rightarrow J$ be the projection. How can I compute the complex $R ...
2
votes
1answer
76 views

local field extension

let $k$ a perfect field and consider the following field $F= k((\omega))$, complete w.r.t a valuation for which $\omega$ is a uniformizer. Consider the field extension $E=F[T]/(T^2-\omega^3)$. Is it ...
0
votes
0answers
72 views

extend a morphism knowing it on etale covering

Let $S$ be a separated scheme. Let $U,J$ be separated schemes over $S$. Assume we can construct, after etale base change $T\rightarrow S$, a map $U_T\rightarrow J_T$ (the sub-T indicates the pulled ...
4
votes
2answers
198 views

isomorphisms of algebraic closures

let $K$ be an algebraically closed field. Consider the algebraic closure $\overline{K(X)}$ of $K(X)$, with $X$ trascendent over $K$. Are there cases in which $\overline{K(X)}\cong K$? where $\cong$ is ...
23
votes
4answers
708 views

Why is it “easier” to work with function fields than with algebraic number fields?

I just bought a copy of Jürgen Neukirch's book Algebraic Number Theory. While browsing through it I found a section titled § 14. Function Fields in chapter I. In it the author describes ...
7
votes
3answers
754 views

What is the operation $\boxtimes$?

Reading papers about $p$-adic analysis and Galois representations, I have found objects like this $D \boxtimes \mathbb{Q}_p$. So my question is what is $\boxtimes$ and how do we read it ?
4
votes
1answer
155 views

$\pi^{tame}(\mathbb{A}^1_k)$ is trivial

Fixed an algebraically closed field of characteristic $p>0$, it is well known the result of the title: $\pi^{tame}(\mathbb{A}^1_k)\simeq 1$. Where the tame fundamental group, in this situation, ...
0
votes
1answer
71 views

theory for contraction maps on curves

do u know any reference for studying the maps $\bar{M}_{i,n}\rightarrow \bar{M}_{j,n-2}$ (moduli space of stable curves with marked points)where the map is given by identifying two marked points? ...
4
votes
5answers
328 views

Derived category and so on

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from ...
6
votes
3answers
319 views

Analogies between Prime Ideals and Knots

While reading this question posted at this link: Subjects studied in number theory i interestingly landed up on this Wikipedia page, and was quite amazed to see the variety of branches opening up. ...
34
votes
1answer
704 views

Geometric intuition behind The Mordell Conjecture

The Mordell Conjecture/Faltings Theorem says roughly that if $K$ is an algebraic number field and $X$ is an algebraic curve defined over $K$ of genus $g >1$ then the set of $K$-rational points ...
2
votes
1answer
315 views

Reduction map from the generic to the special fibre

I have a few basic questions about Liu's [1, Section 10.1.3] description of the reduction map from the closed points of the generic fibre of a proper scheme over a complete DVR to its special fibre. ...
4
votes
2answers
114 views

Bound for number of points on surface over $\mathbb{F}_p$

I know of the bound for the number of points on an elliptic curve over a finite field: $$|\# E(\mathbb{F}_q) - q - 1| < 2\sqrt{q}$$ where this includes the point at infinity. I have been told that ...
7
votes
2answers
361 views

How to Compute Genus

How to compute the genus of $ \{X^4+Y^4+Z^4=0\} \cap \{X^3+Y^3+(Z-tW)^3=0\} \subset \mathbb{P}^3$? We know that the genus of $ \{X^4+Y^4+Z^4=0\} \subset \mathbb{P}^3$ is 3 because the degree is 4. ...
0
votes
1answer
89 views

multiplicity of schemes in positive characteristic

let $K$ a field and $X$ a scheme of finite type over $K$. let $X_1,\dots,X_r$ the irreducible components of $X$ with generic points $\eta_i$. Let $\bar{X}=X\times Spec(\bar{K})$ where $\bar{K}$ is an ...
4
votes
0answers
101 views

On local rings of a normalization

Let X an irreducible singular curve with a singular point $x$. Consider $A$, the normalization of $\mathcal{O}_{X,x}$, and $x_1,\dots,x_r$ the points over $x$ in the normalization of $X$. Why $A\cong ...
0
votes
0answers
143 views

Normalization of a node

how it is described the normalization of $Spec(k[[x,y]]/(xy))$? In particular call this $N$. Why $N$ can be written as the inersection of 2 local rings and the same holds for the maximal ideal (with 2 ...
3
votes
2answers
316 views

Why a smooth surjective morphism of schemes admits a section etale-locally?

Why a smooth surjective morphism of schemes admits a section etale-locally?
4
votes
0answers
91 views

find valuations

consider $f(x)=x^4+5x^3+1$. Let $\alpha$ be a root of this polynomial and consider $K=\mathbb{Q}(\alpha)$. Which are the absolute values on $K$ extending the usual absolute values on $\mathbb{Q}$? Are ...
6
votes
1answer
185 views

places and primes

what does it means that a place divide a prime on an algebraic number field?
3
votes
0answers
103 views

places valuations and so on

I am looking for complete references on places, valuations and so on. In particular I need to understand the meaning of "a finite place does not divide a prime number $\ell$" and what is the Frobenius ...
2
votes
0answers
115 views

Primitive nth roots of unity and moduli of elliptic curves

In various descriptions of the moduli space of elliptic curves with level structures, such as the description of $X_0(N)$ being defined over $Spec\ \mathbb{Z}[1/N]$, the primitive $N^{th}$ roots of ...
15
votes
2answers
1k views

Theories of $p$-adic integration

What is the compelling need for introducing a theory of $p$-adic integration? Do the existing theories of $p$-adic integration use some kind of analogues of Lebesgue measures? That is, do we put a ...
11
votes
3answers
576 views

Importance of determining whether a number is squarefree, using geometry

Despite appearances, this is not a question on computational aspects of number theory. The background is as follows. I once asked a number theorist about what he considered to be the most important ...