Arithmetic of higher-dimensional orbifolds and a mixed Waring problem. (English) Zbl 1508.11041
Summary: We study the density of rational points on a higher-dimensional orbifold \((\mathbb{P}^{n-1},\Delta )\) when \(\Delta\) is a \(\mathbb{Q} \)-divisor involving hyperplanes. This allows us to address a question of S. Tanimoto [Workshop “Rational and integral points via analytic and geometric methods”, Oaxaca (May 27th–June 1st, 2018)] about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy-Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to J. Bourgain et al. [Ann. Math. (2) 184, No. 2, 633–682 (2016; Zbl 1408.11083)] and
T. D. Wooley [Ann. Math. (2) 175, No. 3, 1575–1627 (2012; Zbl 1267.11105); Adv. Math. 294, 532–561 (2016; Zbl 1365.11097); Proc. Lond. Math. Soc. (3) 118, No. 4, 942–1016 (2019; Zbl 1456.11149)].
MSC:
| 11D45 | Counting solutions of Diophantine equations |
| 11P55 | Applications of the Hardy-Littlewood method |
| 14G05 | Rational points |
References:
| [1] | Bourgain, J.; Demeter, C.; Guth, L., Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. Math., 184, 633-682 (2016) · Zbl 1408.11083 · doi:10.4007/annals.2016.184.2.7 |
| [2] | Browning, TD; Van Valckenborgh, K., Sums of three squareful numbers, Experiment. Math., 21, 204-211 (2012) · Zbl 1327.11022 · doi:10.1080/10586458.2011.605733 |
| [3] | Brüdern, J., Sums of squares and higher powers, J. Lond. Math. Soc., 35, 233-243 (1987) · Zbl 0589.10048 · doi:10.1112/jlms/s2-35.2.233 |
| [4] | Campana, F., Fibres multiples sur les surfaces: aspects geométriques, hyperboliques et arithmétiques, Manuscr. Math., 117, 429-461 (2005) · Zbl 1129.14051 · doi:10.1007/s00229-005-0570-5 |
| [5] | Cohen, SD, The distribution of Galois groups and Hilbert’s irreducibility theorem, Proc. Lond. Math. Soc., 3, 227-250 (1981) · Zbl 0484.12002 · doi:10.1112/plms/s3-43.2.227 |
| [6] | Davenport, H., Analytic Methods for Diophantine Equations and Diophantine Inequalities (2005), Cambridge: Cambridge University Press, Cambridge · Zbl 1125.11018 · doi:10.1017/CBO9780511542893 |
| [7] | Franke, J.; Manin, YI; Tschinkel, Y., Rational points of bounded height on Fano varieties, Invent. Math., 95, 421-435 (1989) · Zbl 0674.14012 · doi:10.1007/BF01393904 |
| [8] | Heath-Brown, DR, A new form of the circle method and its application to quadratic forms, J. Reine Angew. Math., 481, 149-206 (1996) · Zbl 0857.11049 |
| [9] | Lang, S.; Weil, A., Number of points of varieties in finite fields, Am. J. Math., 76, 819-827 (1954) · Zbl 0058.27202 · doi:10.2307/2372655 |
| [10] | Pieropan, M., Schindler, D.: Hyperbola method on toric varieties (2020). arXiv:2001.09815 |
| [11] | Pieropan, M., Smeets, A., Tanimoto, S., Várilly-Alvarado, A.: Campana points of bounded height on vector group compactifications. Proc. Lond. Math. Soc · Zbl 1479.11116 |
| [12] | Serre, J.-P.: Lectures on the Mordell-Weil Theorem. Aspects of Mathematics, 3rd edn. Friedr. Vieweg & Sohn, Braunschweig (1997) · Zbl 0863.14013 |
| [13] | Serre, J.-P.: Topics in Galois theory. 2nd edition. Research Notes in Mathematics 1, A. K. Peters, Ltd., Wellesley, MA (2008) |
| [14] | Van Valckenborgh, K., Squareful numbers in hyperplanes, Algebra Number Theory, 6, 1019-1041 (2012) · Zbl 1321.11038 · doi:10.2140/ant.2012.6.1019 |
| [15] | Vaughan, R. C.: The Hardy-Littlewood Method, 2nd ed., Cambridge Tracts in Math. 125, Cambridge Univ. Press, Cambridge (1997) · Zbl 0868.11046 |
| [16] | Wooley, TD, Vinogradov’s mean value theorem via efficient congruencing, Ann. Math., 175, 1575-1627 (2012) · Zbl 1267.11105 · doi:10.4007/annals.2012.175.3.12 |
| [17] | Wooley, TD, The asymptotic formula in Waring’s problem, Int. Math. Res. Not., 2012, 7, 1485-1504 (2012) · Zbl 1267.11104 · doi:10.1093/imrn/rnr074 |
| [18] | Wooley, TD, The cubic case of the main conjecture in Vinogradov’s mean value theorem, Adv. Math., 294, 532-561 (2016) · Zbl 1365.11097 · doi:10.1016/j.aim.2016.02.033 |
| [19] | Wooley, TD, Nested efficient congruencing and relatives of Vinogradov’s mean value theorem, Proc. Lond. Math. Soc., 118, 942-1016 (2019) · Zbl 1456.11149 · doi:10.1112/plms.12204 |
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