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This is an open problem; I personally suspect it cannot happen. More generally let $X$ be a smooth projective variety over a field $k$ which is finitely generated over $\mathbb{Q}$. Then Skorobogatov …

Probably the easiest way to prove this is via flatness. The closure $\bar{x}$ is integral and dominates $S$, thus is flat over $S$ (see Proposition III.9.7 in Hartshorne). The dimension of the fibres …

This answer addresses the second question: "If we assume the fibers of $\pi$ are curves of genus $0$, can we embed $X$ into $\mathbb{P}^2_S$?" The answer to this is also no (providing there is a singu …

Suppose that $C \subset X$ is a smooth projective curve of genus $1$ embedded in a Brauer-Severi surface over a field $k$. We have $C^2 = 9$ since this holds after passing to the algebraic closure, wh …

I suspect that this is unknown in general. I would guess that any method which produces at least one elliptic curve of positive rank should also produce infinitely many of positive rank, which as you …

I will focus attention on smooth projective varieties $X$ over $k$ with $\mathrm{Pic}(X_{\bar{k}})$ a free finitely generated abelian group, as they illustrate all the essential behaviour relevant to …

Surely this behaviour can never happen, but it will be near impossible to prove this. Conjectures of Manin and others predict that there is an asymptotic formula for these functions in many cases, and …

(Upgrading comments to answer.) Counting the number of $D$ for which the equation has a rational solution is a fairly classical problem. This is asymptotic to $c_nD/(\log D)^{1/2}$ for some $c_n > 0$. …

Let $X$ be an integral regular scheme which is proper over $\mathbb{Z}_p$. Assume that the special fibre $X_{\mathbb{F}_p}$ is irreducible and let $k$ be the algebraic closure of $\mathbb{F}_p$ in the …

As explained in the comments, there is no such variety. This is an application of a general set of techniques called "spreading out". You can find a very nice treatment of this in Chapter 3 of the bo …

I hope nobody minds me answering this (very old) question. Me and my collaborators (Barinder Banwait and Francesc Fité) succeeded in completely answering this question in the paper: Del Pezzo surfac …

This is a brief answer; possibly others have different opinions about this. Question 1: The Langlands conjectures gives a correspondence between Galois representations and automorphic forms. So a ver …

This is an attempt at an answer to what I think the question is (please tell me if anything isn't clear). Theorem Let $f:X\to Y $ be a dominant morphism of finite type schemes over $\mathbb{Z}$ with …

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 …

I assume in the question that $C = E$ is an elliptic curve. First your claim that $E(\mathbb{R}) = U(1)$ is false; I mean $E(\mathbb{R})$ can be disconnected. The correct result is that $E(\mathbb{R} …