Generalised divisor sums of binary forms over number fields

Frei, C. and Sofos, E. (2020) Generalised divisor sums of binary forms over number fields. Journal of the Institute of Mathematics of Jussieu, 19(1), pp. 137-173. (doi: 10.1017/s1474748017000469)

Abstract

Estimating averages of Dirichlet convolutions 1 * χ, for some real Dirichlet character χ of fixed modulus, over the sparse set of values of binary forms defined over 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 χ 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 * 1. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than 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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Sofos, Dr Efthymios
• Authors: Frei, C., and Sofos, E.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of the Institute of Mathematics of Jussieu
• Publisher: Cambridge University Press
• ISSN: 1474-7480
• ISSN (Online): 1475-3030
• Published Online: 16 November 2017
• Copyright Holders: Copyright © 2017 Cambridge University Press
• First Published: First published in Journal of the Institute of Mathematics: Jussieu 19(1): 137-173
• Publisher Policy: Reproduced in accordance with the publisher copyright policy

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