450 research outputs found

### Manin's Conjecture for a Singular Sextic del Pezzo Surface

We prove Manin's conjecture for a del Pezzo surface of degree six which has
one singularity of type $\mathbf{A}_2$. Moreover, we achieve a meromorphic
continuation and explicit expression of the associated height zeta function.Comment: 23 pages, 1 figur

### The Hasse principle for lines on diagonal surfaces

Given a number field $k$ and a positive integer $d$, in this paper we
consider the following question: does there exist a smooth diagonal surface of
degree $d$ in $\mathbb{P}^3$ over $k$ which contains a line over every
completion of $k$, yet no line over $k$? We answer the problem using Galois
cohomology, and count the number of counter-examples using a result of
Erd\H{o}s.Comment: 14 page

### Rational points and non-anticanonical height functions

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.Comment: 16 pages; minor corrections; Proceedings of the American Mathematical
Society, 147 (2019), no. 8, 3209-322

### Good reduction of algebraic groups and flag varieties

In 1983, Faltings proved that there are only finitely many abelian varieties
over a number field of fixed dimension and with good reduction outside a given
set of places. In this paper, we consider the analogous problem for other
algebraic groups and their homogeneous spaces, such as flag varieties.Comment: 11 page

### Good reduction of Fano threefolds and sextic surfaces

We investigate versions of the Shafarevich conjecture, as proved for curves
and abelian varieties by Faltings, for other classes of varieties. We first
obtain analogues for certain Fano threefolds. We use these results to prove the
Shafarevich conjecture for smooth sextic surfaces, which appears to be the
first non-trivial result in the literature on the arithmetic of such surfaces.
Moreover, we exhibit certain moduli stacks of Fano varieties which are not
hyperbolic, which allows us to show that the analogue of the Shafarevich
conjecture does not always hold for Fano varieties. Our results also provide
new examples for which the conjectures of Campana and Lang-Vojta hold.Comment: 22 pages. Minor change

### Sieving rational points on varieties

A sieve for rational points on suitable varieties is developed, together with
applications to counting rational points in thin sets, the number of varieties
in a family which are everywhere locally soluble, and to the notion of friable
rational points with respect to divisors. In the special case of quadrics,
sharper estimates are obtained by developing a version of the Selberg sieve for
rational points.Comment: 30 pages; minor edits (final version

### Rational points of bounded height on general conic bundle surfaces

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 or 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.Comment: 35 pages; final versio

### Zero-loci of Brauer group elements on semi-simple algebraic groups

We consider the problem of counting the number of rational points of bounded
height in the zero-loci of Brauer group elements on semi-simple algebraic
groups over number fields. We obtain asymptotic formulae for the counting
problem for wonderful compactifications using the spectral theory of
automorphic forms. Applications include asymptotic formulae for the number of
matrices over Q whose determinant is a sum of two squares. These results
provide a positive answer to some cases of a question of Serre concerning such
counting problems.Comment: 35 pages. Added more details and fixed typos. Final versio

### Manin's conjecture for a singular quartic del Pezzo surface

We prove Manin's conjecture for a split singular quartic del Pezzo surface
with singularity type 2\Aone and eight lines. This is achieved by equipping
the surface with a conic bundle structure. To handle the sum over the family of
conics, we prove a result of independent interest on a certain restricted
divisor problem for four binary linear forms

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