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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

4 votes

Density of polynomials which are soluble with respect to a set of primes

It should be quite possible to obtain an asymptotic formula for this using standard lattice point counting and sieve techniques. First consider the following simpler problem: Let $p$ be a prime and l …
Daniel Loughran's user avatar
5 votes

Rational solutions of $x^2 + y^2 = z (z^2 - 1)$

Consider the corresponding projective surface $$S : x^2w + y^2w = z(z^2 - w^2).$$ No one has mentioned so far that this is surface singular, and singular cubic surfaces are rather special. It has two …
Daniel Loughran's user avatar
8 votes
Accepted

What is the set of possible densities of pointless members in a family of rational curves ov...

Edit: I have clarified a bit more the relationship between conic bundles and quaternion algebras and the relationship to weak approximation, and tidied up some typos. I have myself recently started s …
Daniel Loughran's user avatar
5 votes
Accepted

Birch and Swinnerton-Dyer conjecture in positive characteristic

Edit: This answer addresses an earlier version of the question, where the OP asked whether or not BSD made sense for elliptic curves over finite fields. It also however answers the current question. …
Daniel Loughran's user avatar
2 votes

What are the truly 'global methods' in number theory?

This is quite a brief answer as I am not quite sure your question is appropriately phrased for this site, but one can use the "Brauer-Manin obstruction" to show the non-existence of rational points, e …
Daniel Loughran's user avatar
10 votes

A question on BSD conjecture

Yes. This follows from the fact that BSD is invariant under Weil restriction and isogeny (the Weil restriction of $E/K$ to $\mathbb{Q}$ is isogenous to the product of $E$ with its quadratic twist). …
Daniel Loughran's user avatar
8 votes
0 answers
122 views

Pólya–Vinogradov over number fields

Let $\chi$ be a non-principal Dirichlet character of modulus $q$. The classical Pólya–Vinogradov theorem states that $$\sum_{n \leq x} \chi(n) \leq q^{1/2} \log q.$$ Let now $K$ be a number field of d …
Daniel Loughran's user avatar
12 votes
2 answers
483 views

Twists of cubic threefolds

Let k be a field of characteristic $0$. Let $X$ be a variety over k which is isomorphic to a smooth cubic threefold over $\bar{k}$. Then is $X$ isomorphic to a smooth cubic threefold over $k$ …
Daniel Loughran's user avatar
13 votes

Conics, rational points and probability

Problems of this type are considered by Serre in the paper: Serre - Spécialisation des éléments de $\mathrm{Br}_2(\mathbb{Q}(T_1,\ldots, T_n))$ The case relevant to you is Exemple 4. Here Serre show …
Daniel Loughran's user avatar
4 votes
Accepted

A certain invariant of non-singular algebraic surfaces

I will explain what happens in the case of cubic surfaces. As is well known, any smooth cubic surface over an algebraically closed field contains $27$ lines. For the generic cubic surface over $\mathb …
Daniel Loughran's user avatar
2 votes
Accepted

Equation with norms of cyclic extensions of coprime degrees

Yes, this follows from the fact that the degrees of $\mathbb{K}$ and $\mathbb{L}$ are coprime. The result follows from a simple application of Bezout's identity and the fact that, for an extension $k …
Daniel Loughran's user avatar
6 votes
Accepted

Divergent Series as a topic of research

In some respects the theory of divergent series is still a very important part of number theory. A large part of number theory concerns the study of Dirichlet series $$f(s) = \sum_{n=1}^\infty \frac …
Daniel Loughran's user avatar
13 votes
2 answers
581 views

Sum of Fibonacci sequence evaluated at a Dirichlet character

Let $F_n$ be the Fibonacci sequence and $\chi$ a non-principal primitive Dirichlet character. Does there exist $n$ such that $\chi(F_n) \neq 0,1$? One way to prove this would be to obtain non-trivial …
Daniel Loughran's user avatar
6 votes
Accepted

$H^1$ and fractional ideals group

This vanishing is very different from Hilbert's theorem 90. I will only sketch a proof as it is rather standard and it will be good for you to fill in the details. Your module is a permutation module …
Daniel Loughran's user avatar
5 votes
Accepted

odd degree $0$-cycles and rational points on a quadric hypersurface

Yes. It was a conjecture of Witt, proved by Springer, that a quadric hypersurface, over a field of characteristic not equal to $2$, has a rational point if and only if it has a point over a field exte …
Daniel Loughran's user avatar

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