All Questions
35 questions
1
vote
0
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64
views
Existence of a special uniformizer along a smooth section of a prestable curve
Let $R$ be a complete DVR with fraction field $K$, uniformizer $\pi$ and alg. closed residue field $k$.
Let $f : X\rightarrow \text{Spec }R$ be a prestable model of $\mathbb{P}^1_K$ with a $R$-section ...
- 7,527
1
vote
1
answer
140
views
Specialization of points on the generic fiber in a prestable model of $\mathbb{P}^1$
Let $R$ be a complete DVR with uniformizer $\pi$, fraction field $K$ and residue field $k$. We assume $k$ is algebraically closed.
Let $X$ be a prestable model of $\mathbb{P}^1_K$ over $R$, so $X$ is ...
- 7,527
0
votes
1
answer
160
views
Embedding of symmetric square in Jacobian
Let $C$ be a projective curve defined over a field $K$, and let $C^{(2)}$ and $J$ be its symmetric square and Jacobian, respectively.
There is a natural map $C^{(2)}\hookrightarrow J$, defined as ...
- 2,907
5
votes
1
answer
467
views
Weak Mordell-Weil for EC using Chevalley-Weil theorem
I am reading the book Applications of Diophantine Approximation to Integral Points and Transcendence by Zannier and Corvaja and, after their proof of the Chevalley-Weil theorem, in Example 3.8 they ...
- 233
5
votes
0
answers
225
views
Belyi functions with prescribed image of a given point
$\newcommand{\bP}{\mathbb{P}}\newcommand{\bQ}{\mathbb{Q}}$Definition. A Belyi function is a non-constant rational function $f:\bP_{\bQ}^1\to \bP^1_{\bQ}$ such that the image of any of its critical ...
- 7,377
2
votes
0
answers
242
views
Cartier operator and logarithmic differentials
Let $k$ be an algebraically closed field of characteristic $p$, let $C$ be a curve over $k$ and let $\omega$ be a meromorphic differential form on $C$. If $\omega$ gets mapped to itself by the Cartier ...
- 75
2
votes
1
answer
303
views
Is the reduction of an absolutely irreducible plane curve still irreducible except for the finite number of cases?
Help me please.
Let $k$ be an algebraically closed field (I am mainly interested in $k = \overline{\mathbb{Q}}, \overline{\mathbb{F}_q}$). Consider a plane curve $C \subset \mathbb{A}^2$ of degree $d$ ...
- 1,833
9
votes
1
answer
353
views
The $S$-unit equation for functions on curves
Let $X$ be a smooth projective connected curve over a number field $k$, and let $S \neq \emptyset$ be a finite set of closed points of $X$. The curve $Y = X \setminus S$ is affine, and we denote by $R$...
- 20.8k
5
votes
0
answers
354
views
Modern reference for Andre Weil's 'Sur les courbes algébriques et les variétés qui s'en déduisent'
I'm currently interested in the cardinality of the set of values of a polynomial over a finite field.
I found a paper
Saburo Uchiyama, Sur le Nombre des Valeurs Distinctes d'un Polynome a ...
- 1,013
0
votes
0
answers
80
views
Points on hyperelliptic curves coming from an orbit of an algebraic group
Consider a hyperelliptic curve $C_F$ defined over $\mathbb{P}(1,1,g+1)$ by the equation
$$\displaystyle C_F: z^2 = F(x,y),$$
where $F \in \mathbb{Z}[x,y]$ is a non-singular binary form of degree $2g+...
- 26.9k
30
votes
4
answers
3k
views
Motivation for zeta function of an algebraic variety
If $p$ is a prime then the zeta function for an algebraic curve $V$ over $\mathbb{F}_p$ is defined to be
$$\zeta_{V,p}(s) := \exp\left(\sum_{m\geq 1} \frac{N_m}{m}(p^{-s})^m\right). $$
where $N_m$ is ...
- 901
9
votes
0
answers
216
views
Kronecker's theorem in higher dimension
Recall the following classical theorem of Kronecker: if $P(x) \in \mathbb{Z}[x]$ is a monic irreducible polynomial with all roots on the unit circle $S^1$, then $P(x)$ is a cyclotomic polynomial (and ...
- 20.8k
6
votes
1
answer
704
views
Faltings theorem and number of singularities
The Faltings theorem states that the number of rationals over an algebraic curve is finite if the genus is greater than 1. The genus decreases by increasing the number of singularities. My question is ...
- 1,207
3
votes
1
answer
424
views
Moduli problem of stable nodal curves over the integers
Over an algebraically closed field of characteristic zero, e.g. $\overline{\mathbb{Q}}$, the Deligne-Mumford stack $\overline{\mathcal{M}}_{g,n}$ represents the functor $$\overline{\mathcal{M}}_{g,n}(...
- 7,952
7
votes
0
answers
642
views
Automorphisms of semistable $G$-bundles
Let $X$ be a projective smooth curve over an algebraically closed field $\mathbb k$ of any characteristic. Let also $G$ be a reductive group over $\mathbb k$. (Probably) following Ramanathan, a $G$-...
- 790
1
vote
1
answer
239
views
Local parameters and etale coverings of of elliptic curves
I found some absurd observation which I could not fix by myself. For an elliptic curve $E$ over $\mathbb{Q}$, let $\overline{E}=E\otimes\overline{\mathbb{Q}}$. Every multiplication-by-$n$ map $\...
- 21
15
votes
2
answers
1k
views
Modular forms from counting points on algebraic varieties over a finite field
Suppose we are given some polynomial with integer coefficients, which we regard as carving out an affine variety $E$, for example:
$$ 3x^2y - 12 x^3y^5 + 27y^9 - 2 = 0 \tag{$*$} $$
(We might ...
- 1,821
3
votes
0
answers
394
views
Calculation of Cartier-Manin matrix
Let $\mathbb{F}_q$ be a finite field of characteristic $p$ and let $C$ be a plane projective nonsingular curve over $\mathbb{F}_q$ ,
with function field $K = \mathbb{F}_q(C)$. Let $K^p$ denote the ...
- 2,576
11
votes
1
answer
1k
views
Is Proposition 2.6 in J. Silverman's book Arithmetic of Elliptic Curves correct?
In J. Silverman's book "Arithmetic of Elliptic Curves" Chapter 2 Proposition 2.6 (a) it is considered a non constant morphism $\Phi:C_{1}→C_{2}$ between two smooth curves defined over a perfect field $...
- 675
2
votes
2
answers
440
views
Rational points on towers of curves
Let $\ldots \to X_n \to X_{n-1} \to \ldots \to X_0$ be etale maps between smooth projective curves of genera $g(X_n)>1$, all defined over a fixed number field $K$. By Faltings' Theorem, we know ...
- 16.1k
9
votes
0
answers
649
views
Motivic fundamental group of the moduli space of curves?
Suppose I have a smooth projective family of varieties of varieties over $\mathcal M_g$ - i.e. a universal functor, commuting with deformations, from curves to smooth projective varieties. Can I ...
- 148k
6
votes
2
answers
516
views
Obstruction and rational points on curves
Is etale-Brauer the only obstruction to the existence of rational points on projective plane curves over number fields?
- 11.3k
17
votes
1
answer
3k
views
Why is the section conjecture important?
As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, ...
- 601
9
votes
0
answers
720
views
Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)
Suppose $X/K$ is a curve over a field $K$, which we want to think of as non-algebraically closed, and let $x$ be a point of $X(K)$. The Ceresa cycle is defined as follows; you can embed $X$ in $Jac(X)$...
- 19.2k
1
vote
0
answers
119
views
The minimum genus of a family of degree $12$ algebraic curves which comes from the resultant of two quartic polynomials
Let $f(t)$ be a rational normal cubic curve in $\mathbb{P}^3$ (it is not contained in any plane) and also we assume that this cubic curve passes through two points $(0,0,0)$ and $(1,0,0)$. By an easy ...
- 39
11
votes
0
answers
264
views
What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?
I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that $H^{...
- 148k
4
votes
1
answer
517
views
How to obtain the Period matrix from the Igusa Invariants of a genus two curve?
I am looking for an algorithm to obtain the period matrix tau= (tau1 & tau12\ tau12 & tau2) from the Igusa invariants of a genus two curve. More precisely:
I consider a family of genus two ...
- 41
1
vote
1
answer
307
views
Weierstrass points on modular curves
What is knowns about Weierstrass points on modular curves? Are there any explicit formulas of them, or any information about Weierstrass gaps? I am interested in (compactifications of) the quotients ...
- 5,186
1
vote
2
answers
472
views
Equations of elliptic curves
First part of question I have asked on mathoverflow already: https://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve
1) Let $E(\mathbb{F}_{q^2})$ is elliptic ...
- 2,576
5
votes
1
answer
1k
views
Analogy between Jacobian of curve and Ideal class group
It is excerpt from "Algebraic Geometry Codes Basic Notions"(https://www.google.ru/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCoQFjAA&url=http%3A%2F%2Fwww.math.umass.edu%...
- 2,576
14
votes
3
answers
3k
views
Quadratic reciprocity and Weil reciprocity theorem
I was told that Weil reciprocity theorem (one has two meromorphic function $f,g$ on a complex curve $C$, so $\prod\limits_{x\in C} g(x)^{ord_xf}=\prod\limits_{x\in C}f(x)^{ord_xg} \ $ where $ord_xf$ ...
- 5,053
-1
votes
1
answer
801
views
Genus of algebraic curves with unknown degree
I am not sure if this is a valid question but posting any way:
Say I am over $\mathbb{F}_{p}$ for a prime $p$.
I have a curve of form $x^{2} = f(y)$ where $f(y)$ has an unknown form (and hence ...
18
votes
1
answer
594
views
Does every hyperbolic curve over a finite field have an etale cover with a real Frobenius eigenvalue?
More precisely: let X/F_q be a smooth projective algebraic curve of genus at least 2. Does there always exist a curve Y/F_{q^d} with a finite etale projection Y -> X, such that one of the Frobenius ...
- 19.2k
2
votes
4
answers
617
views
A question on function fields (extending my previous question)
Consider the extension $\mathbb Q(a,b)$ of the field of rationals, where $a$, $b$ are algebraically independent transcendentals. To $\mathbb Q(a,b)$ adjoin the roots of the polynomials $x^5+a^5=1$ and ...
- 161
11
votes
1
answer
705
views
a question on function fields
Consider the transcendental extension Q(t) of the field of rationals.
To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting
field Q(t)[x] is a radical extension of Q(t). Is it true that ...
- 161