The split torsor method for Manin’s conjecture
HTML articles powered by AMS MathViewer
- by Ulrich Derenthal and Marta Pieropan PDF
- Trans. Amer. Math. Soc. 373 (2020), 8485-8524
Abstract:
We introduce the split torsor method to count rational points of bounded height on Fano varieties. As an application, we prove Manin’s conjecture for all nonsplit quartic del Pezzo surfaces of type $\mathbf {A}_{3}+\mathbf {A}_{1}$ over arbitrary number fields. The counting problem on the split torsor is solved in the framework of o-minimal structures.References
- Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR 3307753
- William W. Adams and Philippe Loustaunau, An introduction to Gröbner bases, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR 1287608, DOI 10.1090/gsm/003
- Emil Artin and John Tate, Class field theory, AMS Chelsea Publishing, Providence, RI, 2009. Reprinted with corrections from the 1967 original. MR 2467155, DOI 10.1090/chel/366
- R. de la Bretèche and T. D. Browning, On Manin’s conjecture for singular del Pezzo surfaces of degree four. II, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 3, 579–605. MR 2373960, DOI 10.1017/S0305004107000205
- R. De la Bretèche and T. D. Browning, Manin’s conjecture for quartic del Pezzo surfaces with a conic fibration, Duke Math. J. 160 (2011), no. 1, 1–69. MR 2838351, DOI 10.1215/00127094-1443466
- R. de la Bretèche and T. D. Browning, Binary forms as sums of two squares and Châtelet surfaces, Israel J. Math. 191 (2012), no. 2, 973–1012. MR 3011504, DOI 10.1007/s11856-012-0019-y
- Régis de la Bretèche, Tim Browning, and Emmanuel Peyre, On Manin’s conjecture for a family of Châtelet surfaces, Ann. of Math. (2) 175 (2012), no. 1, 297–343. MR 2874644, DOI 10.4007/annals.2012.175.1.8
- Régis de la Bretèche and Étienne Fouvry, L’éclaté du plan projectif en quatre points dont deux conjugués, J. Reine Angew. Math. 576 (2004), 63–122 (French, with English summary). MR 2099200, DOI 10.1515/crll.2004.091
- David Bourqui, La conjecture de Manin géométrique pour une famille de quadriques intrinsèques, Manuscripta Math. 135 (2011), no. 1-2, 1–41 (French, with English and French summaries). MR 2783385, DOI 10.1007/s00229-010-0403-z
- Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer-Verlag, Berlin, 1995.
- Victor V. Batyrev and Yuri Tschinkel, Manin’s conjecture for toric varieties, J. Algebraic Geom. 7 (1998), no. 1, 15–53. MR 1620682
- Régis de la Bretèche and Gérald Tenenbaum, Sur la conjecture de Manin pour certaines surfaces de Châtelet, J. Inst. Math. Jussieu 12 (2013), no. 4, 759–819 (French, with English and French summaries). MR 3103132, DOI 10.1017/S1474748012000886
- Fabrizio Barroero and Martin Widmer, Counting lattice points and O-minimal structures, Int. Math. Res. Not. IMRN 18 (2014), 4932–4957. MR 3264671, DOI 10.1093/imrn/rnt102
- Antoine Chambert-Loir and Yuri Tschinkel, On the distribution of points of bounded height on equivariant compactifications of vector groups, Invent. Math. 148 (2002), no. 2, 421–452. MR 1906155, DOI 10.1007/s002220100200
- D. F. Coray and M. A. Tsfasman, Arithmetic on singular Del Pezzo surfaces, Proc. London Math. Soc. (3) 57 (1988), no. 1, 25–87. MR 940430, DOI 10.1112/plms/s3-57.1.25
- Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, La descente sur les variétés rationnelles. II, Duke Math. J. 54 (1987), no. 2, 375–492 (French). MR 899402, DOI 10.1215/S0012-7094-87-05420-2
- Ulrich Derenthal, Andreas-Stephan Elsenhans, and Jörg Jahnel, On the factor alpha in Peyre’s constant, Math. Comp. 83 (2014), no. 286, 965–977. MR 3143700, DOI 10.1090/S0025-5718-2013-02772-X
- Ulrich Derenthal, Counting integral points on universal torsors, Int. Math. Res. Not. IMRN 14 (2009), 2648–2699. MR 2520770, DOI 10.1093/imrn/rnp030
- Ulrich Derenthal, Singular del Pezzo surfaces whose universal torsors are hypersurfaces, Proc. Lond. Math. Soc. (3) 108 (2014), no. 3, 638–681. MR 3180592, DOI 10.1112/plms/pdt041
- Kevin Destagnol, La conjecture de Manin pour certaines surfaces de Châtelet, Acta Arith. 174 (2016), no. 1, 31–97 (French). MR 3517531, DOI 10.4064/aa8312-2-2016
- Ulrich Derenthal and Christopher Frei, Counting imaginary quadratic points via universal torsors, Compos. Math. 150 (2014), no. 10, 1631–1678. MR 3269462, DOI 10.1112/S0010437X13007902
- Ulrich Derenthal and Christopher Frei, Counting imaginary quadratic points via universal torsors, II, Math. Proc. Cambridge Philos. Soc. 156 (2014), no. 3, 383–407. MR 3181632, DOI 10.1017/S0305004113000728
- Ulrich Derenthal, Michael Joyce, and Zachariah Teitler, The nef cone volume of generalized del Pezzo surfaces, Algebra Number Theory 2 (2008), no. 2, 157–182. MR 2377367, DOI 10.2140/ant.2008.2.157
- U. Derenthal and D. Loughran, Singular del Pezzo surfaces that are equivariant compactifications, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 377 (2010), no. Issledovaniya po Teorii Chisel. 10, 26–43, 241 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 171 (2010), no. 6, 714–724. MR 2753646, DOI 10.1007/s10958-010-0174-9
- Ulrich Derenthal and Daniel Loughran, Equivariant compactifications of two-dimensional algebraic groups, Proc. Edinb. Math. Soc. (2) 58 (2015), no. 1, 149–168. MR 3333982, DOI 10.1017/S001309151400042X
- Ulrich Derenthal and Marta Pieropan, Cox rings over nonclosed fields, J. Lond. Math. Soc. (2) 99 (2019), no. 2, 447–476. MR 3939263, DOI 10.1112/jlms.12178
- Jens Franke, Yuri I. Manin, and Yuri Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), no. 2, 421–435. MR 974910, DOI 10.1007/BF01393904
- Christopher Frei and Marta Pieropan, O-minimality on twisted universal torsors and Manin’s conjecture over number fields, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 4, 757–811 (English, with English and French summaries). MR 3552013, DOI 10.24033/asens.2295
- H. Heilbronn, Zeta-functions and $L$-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 204–230. MR 0218327
- Edmund Landau, Über Ideale und Primideale in Idealklassen, Math. Z. 2 (1918), no. 1-2, 52–154 (German). MR 1544310, DOI 10.1007/BF01212899
- Yu. I. Manin, Cubic forms, 2nd ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. Algebra, geometry, arithmetic; Translated from the Russian by M. Hazewinkel. MR 833513
- Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
- Emmanuel Peyre, Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J. 79 (1995), no. 1, 101–218 (French). MR 1340296, DOI 10.1215/S0012-7094-95-07904-6
- Emmanuel Peyre, Points de hauteur bornée, topologie adélique et mesures de Tamagawa, J. Théor. Nombres Bordeaux 15 (2003), no. 1, 319–349 (French, with English and French summaries). Les XXIIèmes Journées Arithmetiques (Lille, 2001). MR 2019019
- Marta Pieropan, Torsors and generalized Cox rings for Manin’s conjecture, Ph.D. thesis, Leibniz Universität Hannover, 2015.
- Emmanuel Peyre and Yuri Tschinkel, Tamagawa numbers of diagonal cubic surfaces, numerical evidence, Math. Comp. 70 (2001), no. 233, 367–387. MR 1681100, DOI 10.1090/S0025-5718-00-01189-3
- Michael Rosen, A generalization of Mertens’ theorem, J. Ramanujan Math. Soc. 14 (1999), no. 1, 1–19. MR 1700882
- J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216
- Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd ed., Graduate Studies in Mathematics, vol. 163, American Mathematical Society, Providence, RI, 2015. Translated from the 2008 French edition by Patrick D. F. Ion. MR 3363366, DOI 10.1090/gsm/163
- Eckart Viehweg, Vanishing theorems, J. Reine Angew. Math. 335 (1982), 1–8. MR 667459, DOI 10.1515/crll.1982.335.1
- A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), no. 4, 1051–1094. MR 1398816, DOI 10.1090/S0894-0347-96-00216-0
Additional Information
- Ulrich Derenthal
- Affiliation: Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Welfengarten 1, 30167 Hannover, Germany
- MR Author ID: 776744
- Email: derenthal@math.uni-hannover.de
- Marta Pieropan
- Affiliation: Utrecht University, Mathematical Institute, Budapestlaan 6, 3584 CD Utrecht, the Netherlands
- MR Author ID: 1166273
- Email: m.pieropan@uu.nl
- Received by editor(s): July 22, 2019
- Received by editor(s) in revised form: February 5, 2020
- Published electronically: September 29, 2020
- Additional Notes: The first author was partly supported by grant DE 1646/4-2 of the Deutsche Forschungsgemeinschaft. Some of this work was done while he was on sabbatical leave at the University of Oxford.
The second author was partly supported by grant ES 60/10-1 of the Deutsche Forschungsgemeinschaft. - © Copyright 2020 by the authors
- Journal: Trans. Amer. Math. Soc. 373 (2020), 8485-8524
- MSC (2010): Primary 11D45; Secondary 11G35, 14G05
- DOI: https://doi.org/10.1090/tran/8133
- MathSciNet review: 4177266