abstract: The proofs of Theorem 1.1 and Theorem 1.5\,(2) in the authors' paper \emph{The Hasse norm principle for abelian extensions} are incorrect. We point out the mistakes and provide correct proofs, using techniques of the original paper.

How many natural numbers below X can be written as a sum of k units of the ring of integers of a given number field? We give the asymptotics as X gets large for quadratic number fields. This solves a problem of Jarden and Narkiewicz from 2007 for quadratic number fields.

The proofs of Theorem 1.1 and Theorem 1.5(2) in the authors' paper 'The Hasse norm principle for abelian extensions' are incorrect. We point out the mistakes and provide correct proofs, using techniques of the original paper.

We study the quantitative behaviour of genus numbers of abelian extensions of number fields with given Galois group. We prove an asymptotic formula for the average value of the genus number and show that any given genus number appears only 0%$0\%$ of the time.

Given a number field $K$, a finite abelian group $G$ and finitely many elements $\alpha_1,\ldots,\alpha_t\in K$, we construct abelian extensions $L/K$ with Galois group $G$ that realise all of the elements $\alpha_1,\ldots,\alpha_t$ as norms of elements in $L$. In particular, this shows existence of such extensions for any given parameters. Our approach relies on class field theory and a recent formulation of Tate's characterisation of the Hasse norm principle, a local-global principle for norms. The constructions are sufficiently explicit to be implemented on a computer, and we illustrate them with concrete examples.

We study the distribution of extensions of a number field k with fixed abelian Galois group G, from which a given finite set of elements of k are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for 100% of G-extensions of k, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.

We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for 100% of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.

The first purpose of our paper is to show how Hooley's celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy-Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series. In particular, using results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation as an Euler product and prove that this does not extend to a complete factorisation.

. We study the distribution of extensions of a number field k with fixed abelian Galois group G , from which a given finite set of elements of k are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for 100% of G -extensions of k , when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.

We prove upper bounds for the average size of the ℓ -torsion Cl _K[ℓ ] of the class group of K , as K runs through certain natural families of number fields and ℓ is a positive integer. We refine a key argument, used in almost all results of this type, which links upper bounds for Cl _K[ℓ ] to the existence of many primes splitting completely in K that are small compared to the discriminant of K . Our improvements are achieved through the introduction of a new family of specialised invariants of number fields to replace the discriminant in this argument, in conjunction with new counting results for these invariants. This leads to significantly improved upper bounds for the average and sometimes even higher moments of Cl _K[ℓ ] for many families of number fields K considered in the literature, for example, for the families of all degree- d -fields for d∈{2,3,4,5} (and non- D_4 if d=4 ). As an application of the case d=2 we obtain the best upper bounds for the number of D_p -fields of bounded discriminant, for primes p>3 .

Estimating averages of Dirichlet convolutions $1\ast \unicode[STIX]{x1D712}$, for some real Dirichlet character $\unicode[STIX]{x1D712}$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin’s conjecture for Châtelet surfaces. We introduce a far-reaching generalisation of this problem, in particular replacing $\unicode[STIX]{x1D712}$ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin’s conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number.

A conjecture of Batyrev and Manin predicts the asymptotic behaviour of rational points of bounded height on smooth projective varieties over number fields. We prove some new cases of this conjecture for conic bundle surfaces equipped with some non-anticanonical height functions. As a special case, we verify these conjectures for the first time for some smooth cubic surfaces for height functions associated to certain ample line bundles.

We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms, this improves significantly upon an earlier result of Dey and Thangadurai.

For all positive integers ℓ , we prove non-trivial bounds for the ℓ -torsion in the class group of K , which hold for almost all number fields K in certain families of cyclic extensions of arbitrarily large degree. In particular, such bounds hold for almost all cyclic degree- p -extensions of F , where F is an arbitrary number field and p is any prime for which F and the p th cyclotomic field are linearly disjoint. Along the way, we prove precise asymptotic counting results for the fields of bounded discriminant in our families with prescribed splitting behavior at finitely many primes.

We prove upper bounds for the average size of the l-torsion ClK [l] of the class group of K, as K runs through certain natural families of number fields and l is a positive integer. We refine a key argument, used in almost all results of this type, which links upper bounds for ClK [l] to the existence of many primes splitting completely in K that are small compared to the discriminant of K. Our improvements are achieved through the introduction of a new family of specialised invariants of number fields to replace the discriminant in this argument, in conjunction with new counting results for these invariants. This leads to significantly improved upper bounds for the average and sometimes even higher moments of ClK [l] for many families of number fields K considered in the literature, for example, for the families of all degree-d-fields for d ∈ {2, 3, 4, 5} (and non-D4 if d = 4). As an application of the case d = 2 we obtain the best upper bounds for the number of Dp-fields of bounded discriminant, for primes p > 3.

We study the distribution of abelian extensions of bounded discriminant of a number field k which fail the Hasse norm principle. For example, we classify those finite abelian groups G for which a positive proportion of G-extensions of k fail the Hasse norm principle. We obtain a similar classification for the failure of weak approximation for the associated norm one tori. These results involve counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which we achieve using tools from harmonic analysis, building on work of Wright.

A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds for a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.

Consider a system of polynomials in many variables over the ring of integers of a number field K. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety X subset of P-k(m).; satisfies the Hasse principle, weak approximation, and the Manin Peyre-conjecture if only its dimension is large enough compared to its degree. This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where K = Q. Our main tool is Skinner's number field version of the Hardy-Littlewood circle method. As a by-product, we point out and correct an error in Skinner's treatment of the singular integral.

Manin's conjecture predicts the distribution of rational points on Fano varieties. Using explicit parameterizations of rational points by integral points on universal torsors and lattice-point-counting techniques, it was proved for several specific varieties over $\mathbb{Q}$, in particular del Pezzo surfaces. We show how this method can be implemented over arbitrary number fields $K$, by proving Manin's conjecture for a singular quartic del Pezzo surface of type $\mathbf{A}_3+\mathbf{A}_1$. The parameterization step is treated in high generality with the help of twisted integral models of universal torsors. To make the counting step feasible over arbitrary number fields, we deviate from the usual approach over $\mathbb{Q}$ by placing higher emphasis on the geometry of numbers in the framework of o-minimal structures.

We prove that any smooth cubic surface defined over any number field satisfies the lower bound predicted by Manin's conjecture possibly after an extension of small degree.