450 research outputs found

    Manin's Conjecture for a Singular Sextic del Pezzo Surface

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    We prove Manin's conjecture for a del Pezzo surface of degree six which has one singularity of type A2\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

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    Given a number field kk and a positive integer dd, in this paper we consider the following question: does there exist a smooth diagonal surface of degree dd in P3\mathbb{P}^3 over kk which contains a line over every completion of kk, yet no line over kk? 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

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    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

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    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

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    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

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    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

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    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

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    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

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    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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