Tagged Questions

5
votes
1answer
49 views

Why do varieties with torsion canonical sheaf have finite etale covers with trivial canonical sheaf

Let $B$ be a variety with torsion canonical sheaf, i.e., $\omega^{\otimes n}_B \cong \mathcal O_B$ for some $n>0$. Then, why does there exist a finite (etale?) morphism $X\to B$ such that $K_X$ is ...
6
votes
0answers
51 views

Families of curves over number fields

Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive ...
3
votes
1answer
44 views

Torsors under elliptic curves splitting over the same fields

I have a question somewhat related to my last question. Suppose $C$ and $C'$ are two genus $1$ curves (smooth, projective, geom conn.) over a perfect field $k$ with no $k$-rational points and that $C$ ...
6
votes
1answer
65 views

If $E/\mathbf Q$ is an elliptic curve and $n$ is odd, then the $n$-torsion $E(\mathbf Q)[n]$ is cyclic; elementary proof?

I know that this follows from the existence and non-degeneracy of the Weil pairing. A consequence of the existence of the Weil pairing is that, if the whole $n$-torsion is defined over $\mathbf Q$, ...
34
votes
3answers
395 views

The resemblance between Mordell's theorem and Dirichlet's unit theorem

The first one states that if $E/\mathbf Q$ is an elliptic curve, then $E(\mathbf Q)$ is a finitely generated abelian group. If $K/\mathbf Q$ is a number field, Dirichlet's theorem says (among other ...
3
votes
0answers
31 views

What are the easiest surfaces of general type

The "easiest type" of curves of general type are those of genus two. In this case $\chi(X,\mathcal O_X) = -1$ and $\deg K = 2$, where $K$ is the canonical sheaf. I'm a bit lost when it comes to ...
5
votes
1answer
69 views

Ramification indices and residue degrees of a finite Galois extension

Let $A$ be a dvr with fraction field $K$ of characteristic zero. Let $L/K$ be a finite Galois extension and let $B$ be the integral closure of $A$ in $L$. For a prime $b$ of $B$, let $e_b$ be its ...
5
votes
1answer
78 views

Definition of tamely ramified

I think I can show that the following definitions of "tamely ramified" coincide. I thought it would be good to be sure. Sorry for the easy questions. Let $O_K$ be a dvr with maximal ideal $\mathfrak ...
6
votes
1answer
134 views

Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused)

Let $O_K$ be a dvr with fraction field $K$. Let $L/K$ be a tamely ramified finite Galois extension. Then, Abhyankar's Lemma implies that there exists a finite Galois extension $K^\prime/K$ such that ...
6
votes
1answer
119 views

Are there infinitely many pairs of rational numbers such that…

Are there infinitely many pairs of rational numbers $(a,b)$ such that $a^3+1$ is not a square in $\mathbf{Q}$, $b^3+2$ is not a square in $\mathbf{Q}$ and $b^3+2 = x^2(a^3+1)$ for some $x$ in ...
2
votes
1answer
80 views

Primes of good reduction for varieties

Suppose I have a (smooth, projective) variety $X/\mathbb{Q}$. Is the notion of a primes of bad/good reduction well-defined from this data? Motivation and attempt at an answer: The question should be ...
2
votes
1answer
105 views

Can we descend field extensions of prime degree of number fields to number fields of the same degree

Let $K$ be a number field and let $p$ be a prime number. Let $L$ be a degree $p$ field extension of $K$. Does there exist a degree $p$ field extension $M$ of $\mathbf{Q}$ such that ...
14
votes
3answers
501 views

Important papers in arithmetic geometry and number theory

Having been inspired by this question I was wondering, what are some important papers in arithmetic geometry and number theory that should be read by students interested in these fields? There is a ...
3
votes
1answer
114 views

Very special rational points on curves over number fields

For some reason, I'm convinced the answer to the following question should be (obviously) negative, but I can't come up with a good reason. Does there exist a number field $K$, a smooth projective ...
4
votes
1answer
135 views

Does every curve over a number field have infinitely many rational functions of fixed degree

Let $X$ be a curve over a number field $K$ of genus $g\geq 2$. Does there exist an integer $d$ such that $X$ has infinitely many rational functions (i.e., finite morphisms $f:X\to \mathbf{P}^1_K$) of ...
6
votes
1answer
341 views

Translation for EGA/SGA

People often recommend Grothendieck's EGA (Elements de Geometrie Algebrique) and SGA (seminaire de geometrie algebrique) as a good reference for learning arithmetic geometry. However, as the title ...
1
vote
0answers
116 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 ...
4
votes
1answer
116 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 ...
4
votes
2answers
127 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 ...
5
votes
3answers
247 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. ...
32
votes
1answer
526 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 ...
4
votes
2answers
96 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 ...
4
votes
0answers
78 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
154 views

places and primes

what does it means that a place divide a prime on an algebraic number field?
10
votes
2answers
690 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 ...