Average Bateman--Horn for Kummer polynomials

For any $r \in \mathbb{N}$ and almost all $k \in \mathbb{N}$ smaller than $x^r$, we show that the polynomial $f(n) = n^r + k$ takes the expected number of prime values as $n$ ranges from 1 to $x$. As a consequence, we deduce statements concerning variants of the Hasse principle and of the integral Hasse principle for certain open varieties defined by equations of the form $N_{K/\mathbb{Q}}(\mathbf{z}) = t^r +k \neq 0$ where $K/\mathbb{Q}$ is a quadratic extension. A key ingredient in our proof is a new large sieve inequality for Dirichlet characters of exact order $r$.Comment: V2: Minor correction

Degrees of closed points on diagonal-full hypersurfaces

Let $k$ be any field. Let $X \subset \mathbb{P}_k^N$ be a diagonal-full degree $d$ hypersurface, where $d$ is an odd prime. We prove that if $X(K) \neq \emptyset$ for some extension $K/k$ with $n:=[K:k]$ prime and $gcd(n,d)=1$, then $X(L) \neq \emptyset$ for some extension $L/k$ with $gcd([L:k], nd)=1$ and $[L:k] \leq nd-n-d$. Moreover, if a $K$-solution is known explicitly, then we can compute $L/k$ explicitly as well. When $n$ or $d$ is not prime, we can still say something about the possible values of $[L:k]$. As an example, we improve upon a theorem by Coray on smooth cubic surfaces $X \subset \mathbb{P}^3_k$, in the case when $X$ is diagonal-full, by showing that if $X(K) \neq \emptyset$ for some extension $K/k$ with $gcd([K:k], 3)=1$, then $X(L) \neq \emptyset$ for some $L/k$ with $[L:k] \in \{1, 10\}$.Comment: Comments welcome

Arithmetic of rational points and zero-cycles on products of Kummer varieties and K3 surfaces

Let k be a number field. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of k to that of zero-cycles over k for Kummer varieties over k. For example, for any Kummer variety X over k, we show that if the Brauer-Manin obstruction is the only obstruction to the Hasse principle for rational points on X over all finite extensions of k, then the (2-primary) Brauer-Manin obstruction is the only obstruction to the Hasse principle for zero-cycles of any given odd degree on X over k. We also obtain similar results for products of Kummer varieties, K3 surfaces and rationally connected varieties

Campana points on diagonal hypersurfaces

We construct an integral model for counting Campana points of bounded height on diagonal hypersurfaces of degree greater than one, and give an asymptotic formula for their number, generalising work by Browning and Yamagishi. The paper also includes background material on the theory of Campana points on hyperplanes and previous results in the field.Comment: 19 page