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Let $k$ be a field (I'm mainly interested in the case where $k$ is a number field, however results for other fields would be interesting), and $X$ a smooth projective variety over $k$. By a zero cyc …

Let $k$ be a finite abelian extension of $\mathbb{Q}$. Class field theory states that $k$ corresponds to some open subgroup of finite index $U_k \subset \mathbb{A}_{\mathbb{Q}}^*/ \mathbb{Q}^*$ where …

It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what th …

I recently came across an interesting phenomenon which confused me slightly, concerning integral points on varieties. For example, consider $X = \mathbb{A}_{\mathbb{Z}}^{n+1} \setminus \{0\}$, affine …

By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$. Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite f …

Any such matrix $M$ would give rise to an automorphism of the cubic surfaces $$a^3 + b^3 = c^3 \pm d^3 \quad \subset \mathbb{P}^3.$$ These are both just different ways of writing the Fermat cubic sur …

I am looking for a version of Ehresmann's theorem for analytic manifolds over the $p$-adic numbers $\mathbb{Q}_p$ or, more generally, local fields. I follow the conventions from Serre's book "Lie alge …

This is not possible in any meaningful way. In fact the variety you describe defines a smooth cubic threefold $X$ in $\mathbb{P}^4$. By a famous theorem of Clemens and Griffiths these are not even rat …

Recall that the radical of an integer $n$ is defined to be $\operatorname{rad}(n) = \prod_{p \mid n } p$. For a paper, I need the result that $$\sum_{n \leq x} \frac{1}{\operatorname{rad}(n)} \ll_\v …

The heuristic fails for precisely the reason you state; there is a parametric family of solutions which makes it fail. People often use heuristic arguments to predict the number of integral/rational …

Let $K/\mathbb{Q}$ be a number field. We say that a rational prime $p$ splits in $K$ if there exists a prime $\mathfrak{p}$ of $K$ above $p$ of interia degree $1$. Is a number field $K$ uniq …

It seems like you want to discover analytic number theory. There is a lot of it. A good comprehensive modern book I would recommend is Iwaniec, Kowalski - Analytic number theory. Example areas with …

This question has a false premise; there is no such thing as a "formula for a general solution" in this case, exactly for the reason that user sdr describes. I interpret the question as follows. The O …

Let $E: y^2 = x^3 + ax + b$ be an elliptic curve over $\mathbb{Q}$ with $a,b \in \mathbb{Z}$. Recall that any rational point $P = (x,y)$ can be written uniquely as $P = (u/d^2, v/d^3)$ with $u,v,d \in …

In Lemma 2 of [1], Heath-Brown proves the following (I state a simplified version of a more general result): Let $\Lambda \subset \mathbb{Z}^2$ be a lattice of determinant $d(\Lambda)$. Then $$\ …