0
votes
1answer
112 views

Arithmetic property of a surface of general type

In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general ...
3
votes
3answers
458 views

Non existence of cyclic infinite linear algebraic groups

Let $G$ be a linear algebraic group defined over some algebraically closed field $\mathbb{K}$ and also over some subfield $k\subset \mathbb{K}$. There is thus a natural group structure on the set of ...
4
votes
0answers
118 views

Continuity of the Hilbert pairing

I would like to know if the Kummer pairing (or the analogue of the Hilbert Symbol) for a one dimensional group defined over the ring of integers of a higher-dimensional local field is continuous (with ...
11
votes
4answers
447 views

Analogy between the nodal cubic curve $y^2=x^3+x^2$ and the ring $\mathbb{Z}[\sqrt{-3}]$?

I'm trying to motivate a bit of algebraic geometry in an abstract algebra course (while simultaneously trying to learn a bit of algebraic geometry), and I thought that it might be nice to present an ...
2
votes
1answer
240 views

Mordell-Weil and finiteness of rational points

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order, that is, $\mathrm{End}_K(E)\cong \mathcal{O}_K$. Suppose the class number of $K$ is equal to $1$. Let ...
2
votes
0answers
148 views

Lang's preprint “Cyclotomic points, very anti-canonical varieties, and quasi-algebraic closure”

I am trying to find the following preprint of Serge Lang, which supposedly discusses his C1 conjecture: "Cyclotomic points, very anti-canonical varieties, and quasi-algebraic closure". I have not ...
1
vote
1answer
207 views

Rational points of non-rational curves

An algebraic curve (in this question) is the zero set   $C = f^{-1}(X\ Y)$ of any polynomial   $f\in\mathbb R[X\ Y]$;   we say then that   $f$   represents   $C$.   ...
1
vote
0answers
54 views

Isogenies in multidiensional formal groups

Let $K/\mathbb{Q}_p$ be a local field, $A$ the ring of integers of K, $\pi$ a uniformizer element for $A$, $F$ an n-dimensional formal group with coefficients in $A$ and $f$ an endomorphism of $F$. ...
3
votes
1answer
182 views

Modular Functions with Rational Fourier Expansions

I have been reading the paper of Cox, McKay and Stevenhagen "Principal Moduli and Class Fields", http://arxiv.org/pdf/math/0311202v1.pdf, and I have a question regarding the nature of the function ...
3
votes
1answer
117 views

How to estimate a local hilbert samuel funcion

Let $X$ be a reduced hypersurface in the projective variety $\mathbb{P}^n(K)$, where $K$ is a number field. Select $\xi$ is a $F_{\mathfrak{p}}$-rational point of $X$ where $\mathfrak{p}$ is a prime ...
5
votes
1answer
396 views

Analogy between Jacobian of curve and Ideal class group

It is excerpt from "Algebraic Geometry Codes Basic ...
10
votes
1answer
789 views

Are overlaps among {algebraic geometry, arithmetic geometry, algebraic number theory} growing?

From a naive outsider's viewpoint, just watching the MO postings in those three fields scroll by, and hearing of breakthroughs in the news, it appears there might be increasing overlap among the ...
15
votes
1answer
439 views

Is there a known example of a curve X of genus > 1 over Q such that we know the number of points of X over the n-th cyclotomic field, for every n?

By Falting's theorem, these numbers are of course finite. Is there an example where we can explicitly compute them for every $n$? Thank you!
1
vote
0answers
83 views

points in $V(\bar K \otimes_{\bar Q} \bar L)$ rational over tensor product of fields

Let V be a variety over a number field, and let K and L be two algebraically closed What is known about the points of $V(\bar K \otimes_{\bar Q} \bar L )$ ? Are there results claiming that points in ...
3
votes
0answers
116 views

P-adic Weierstrass Lemma for several variables

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...
14
votes
1answer
680 views

Principal maximal ideals in Z[x]/(F)

Is there some irreducible $F \in \mathbb{Z}[x]$ such that $\mathbb{Z}[x]/(F)$ has no principal maximal ideal? Equivalently, is it possible that the $1$-dimensional integral domain $\mathbb{Z}[x]/(F)$ ...
6
votes
1answer
271 views

Rational points on surfaces of general type

The weak Lang conjecture asserts that rational points on a variety of general type defined over $\mathbb{Q}$ are not Zariski dense (same replacing $\mathbb{Q}$ with a number field). This one is proved ...
1
vote
1answer
182 views

Functional equations of zeta functions over global fields

The functional equations for Dedekind zeta functions (zeta functions attached to rings of integers in algebraic number fields) come from functional equations of theta functions like $\sum_{n \in ...
2
votes
2answers
163 views

An expression for the function $f_e$ that appears in the Weil Pairing

Let $K$ be a local field and $E/K$ an elliptic curve such that the set of $N$-torsion points, $E[N]$, is contained in $E(K)$. For $e$ in $E[N]$, I am interested in finding and expression for the ...
4
votes
3answers
354 views

Orders of Number Fields

Let $K$ be a number field over $\mathbb{Q}$ of degree $n$, and $\mathcal{O} \subset \mathcal{O}_K$ an order. $\textbf{Questions:}$ $\newcommand{\Spec}{\textrm{Spec }}$ $\newcommand{\cO}{\mathcal{O}}$ ...
6
votes
2answers
397 views

Jacobians defined over smaller fields

Let $L/K$ be an extension of number fields. Let $X$ be a curve over $L$ which can not be defined over $K$. Let $J(X)$ be the Jacobian of $X$ over $L$. In general, the Jacobian $J(X)$ probably ...
6
votes
1answer
610 views

$\ell$-adic Weil cohomology theory

I have a reference or counterexample request. Suppose $k$ is a field and $\ell\neq char(k)$. There are several common references that show that $H^i_{et}(-, \mathbb{Q}_\ell )$ is a Weil cohomology ...
2
votes
3answers
629 views

Are all Finite Subsets of Affine n-space Algebraic sets, and related question [closed]

For an algebraicaly closed field $k$ are all finite subsets of Affine $n$-space $A^{n}\left(k\right)$ algebraic sets (here for $n>1$), and if so, for a given finite set $X\subset ...
4
votes
0answers
259 views

What is the shape of the zeta function of a singular hypersurface?

So let $X$ be a projective hypersurface inside $\mathbb{P}_{\mathbb{Z}}^n$ of degree $d$. Assume that (a) $X(\mathbb{C})$ and $X(\overline{\mathbb{F}}_p)$ are irreducible, (b) and that ...
1
vote
1answer
279 views

Centralizer of elliptic elements in $GL(2)$

Consider a global field $F$ and the group $\Gamma =GL(2,F)$. An element $\gamma \in \Gamma$ is called elliptic, if its eigenvalues do not lie in $F$. Now consider a completion $F_v$ of $F$ and $G_v = ...
0
votes
3answers
1k views

Where Can i find the lecture Videos of BSD 2011

i recently heard that there was a conference on Birch and Swinnerton dyer conjecture Held at Cambridge on May 4 until May 6, the main theme is "The conference marks the 50th anniversary of the ...
27
votes
3answers
2k views

On what kind of objects do the Galois groups act?

I am neither number theorist nor algebraic geometer. I am wondering whether Galois groups of number fields (say the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which ...
2
votes
1answer
454 views

Endomorphism ring of the adeles and ideles?

What are the (cont.) endomorphisms resp. automorphisms of the adeles (for a given global field) 1) as a topological abelian group and 2) as a topological ring? 3) What are the endomorphisms and the ...
13
votes
4answers
2k views

Conceptualizing Weil Pairing for elliptic curves ( and number fields)

There are two explanations in Silverman ( Arithmetic of Elliptic Curves), one in exercises developing the Weil reciprocity law ( for algebraic curves) and then generalizing, and then there is a ...
7
votes
1answer
721 views

Parametrization of 2-dimensional torus

The units with norm $+1$ in a pure cubic number field $K$ generated by a cube root of $m = ab^2$, where $a$ and $b$ are coprime and squarefree integers, correspond to integral points on the torus $$ ...
19
votes
1answer
892 views

Weil Conjectures for Number Fields

Let $K$ be a number field with integral basis $\{\omega_1,\ldots,\omega_n\}$. The affine variety $A_K$ defined by $$ N_{K/\{\mathbb Q}}(X_1 \omega_1 + \ldots + X_n \omega_n) = 1 $$ is an algebraic ...
5
votes
1answer
232 views

Parametrization of unit varieties

Let $K$ be a number field with integral basis $\{\omega_1,\ldots,\omega_n\}$. Then $$ \Phi(X_1, \ldots, X_n) = N_{K/{\mathbb Q}}(\omega_1 X_1 + \ldots + \omega_n X_n) $$ is a homogeneous polynomial of ...
2
votes
1answer
601 views

A unique zero of a system of polynomials is a zero of a finite system.

Suppose $p$ is a point in $\mathbb{R}^n$ so that among the set $S$ of polynomials in $\mathbb{Z}[x_1,\ldots,x_n]$ which equal zero at $p$, $p$ is the only point in some neighborhood of $p$ at which ...
5
votes
2answers
1k views

Is there a Riemann-Roch for smooth projective curves over an arbitrary field?

Let $X$ be a smooth projective curve over a field $k$. We let $\omega$ be the canonical line bundle of $X$ and we denote by $F$ the field of $k$-valued rational functions on $X$. (1) When $k$ is ...
12
votes
3answers
624 views

Frobenius elements from the point of view of étale fundamental groups

The goal of this question is to find a "geometric" definition of Frobenius element in $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Here are two definitions that don't work, but that should help ...
6
votes
2answers
715 views

“Bad” reduction of Shimura curves via dual graphs

I have the following naive (and inexpert) question about the reduction of Shimura curves at primes dividing the discriminant of the underlying quaternion algebra. It requires some background to ...
4
votes
0answers
263 views

What to call the following variant of tame ramification

Suppose that $R \subseteq S$ is a generically separable extension of 1-dimensional normal domains (you can assume that $R$ is local if you'd like) of equal-characteristic $p > 0$ (for simplicity, ...
5
votes
4answers
2k views

Elliptic curves over finite fields

I have basic questions about elliptic curves over finite fields. Where to find general references? Hartshorne for instance restricts to algebraically closed ground fields. Over an arbitrary field ...
2
votes
1answer
637 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 ...
14
votes
4answers
717 views

Etale coverings of certain open subschemes in Spec O_K

Although my number theory is really weak, I'm trying to understand the notion of etale coverings in this context. I think this could provide a very interesting point of view. Let $U$ be an open ...
4
votes
1answer
621 views

2d Weil conjecture

Does there exist a two variable analogue of the Weil conjecture? What I mean is that the usual Weil involves a one-variable zeta-function which you get by using numbers $V_n = V ( GF(p^n))$ of points ...
6
votes
1answer
442 views

Can algebraic number fields be generalized in a similar way to function fields in 1 variable over a finite field?

Global fields consist of finite extensions of $\mathbb{Q}$ (algebraic number fields) and finite extensions of $\mathbb{F}_q(x)$ (function fields in 1 variable over a finite field). The latter are ...
13
votes
1answer
475 views

components of E[p], E universal in char p.

I have just realised that a group scheme I've known and loved for years is probably a bit wackier than I'd realised. In this question, in Charles Rezk's answer, I erroneously claim that his ...
9
votes
1answer
397 views

How to find examples of non-trival kernel of maps between Brauer groups Br(R) -> Br(K)

Background/Motivation: The facts about the Brauer groups I will be using are mainly in Chapter IV of Milne's book on Etale cohomology (unfortunately it was not in his online note). Let $R$ be a ...