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Counting imaginary quadratic points via universal torsors, II

Published online by Cambridge University Press:  15 January 2014

ULRICH DERENTHAL
Affiliation:
Mathematisches Institut, Ludwig-Maximilians-Universität MtünchenTheresienstr. 39, 80333 Mtünchen, Germany e-mail: ulrich.derenthal@math.lmu.de, christopher.frei@math.lmu.de
CHRISTOPHER FREI
Affiliation:
Mathematisches Institut, Ludwig-Maximilians-Universität MtünchenTheresienstr. 39, 80333 Mtünchen, Germany e-mail: ulrich.derenthal@math.lmu.de, christopher.frei@math.lmu.de

Abstract

We prove Manin's conjecture for four singular quartic del Pezzo surfaces over imaginary quadratic number fields, using the universal torsor method.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2014 

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References

REFERENCES

[BB07]de la Bretèche, R. and Browning, T. D.On Manin's conjecture for singular del Pezzo surfaces of degree 4. I. Michigan Math. J. 55 (1) (2007), 5180.Google Scholar
[BD09]Browning, T. D. and Derenthal, U.Manin's conjecture for a quartic del Pezzo surface with A 4 singularity. Ann. Inst. Fourier (Grenoble) 59 (3) (2009), 2311265.CrossRefGoogle Scholar
[BM90]Batyrev, V. V. and Manin, Yu. I.Sur le nombre des points rationnels de hauteur borné des variétés algébriques. Math. Ann. 286 (1–3) (1990), 2743.Google Scholar
[BT95]Batyrev, V. V. and Tschinkel, Yu.Rational points of bounded height on compactifications of anisotropic tori. Int. Math. Res. Not. IMRN (12) (1995), 591635.Google Scholar
[BT98a]Batyrev, V. V. and Tschinkel, Yu.Manin's conjecture for toric varieties. J. Algebraic Geom. 7 (1) (1998), 1553.Google Scholar
[BT98b]Batyrev, V. V. and Tschinkel, Yu.Tamagawa numbers of polarized algebraic varieties. Astérisque (251) (1998), 299340. Nombre et répartition de points de hauteur bornée (Paris, 1996).Google Scholar
[CLT02]Chambert–Loir, A. and Tschinkel, Yu.On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math. 148 (2) (2002), 421452.Google Scholar
[Der06]Derenthal, U.Singular Del Pezzo surfaces whose universal torsors are hypersurfaces. Proc. Lond. Math. Soc. (3), to appear, arXiv:math.AG/0604194 (2006).Google Scholar
[Der09]Derenthal, U.Counting integral points on universal torsors. Int. Math. Res. Not. IMRN (14) (2009), 26482699.Google Scholar
[DF13]Derenthal, U. and Frei, C. Counting imaginary quadratic points via universal torsors. Compos. Math., to appear, arXiv:1302.6151, (2013).Google Scholar
[DL10]Derenthal, U. and Loughran, D.Singular del Pezzo surfaces that are equivariant compactifications. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 377 (Issledovaniya po Teorii Chisel. 10) 241 (2010), 2643.Google Scholar
[DL12]Derenthal, U. and Loughran, D. Equivariant compactifications of two-dimensional algebraic groups. Proc. Edinb. Math. Soc., to appear, arXiv:1212.3518 (2012).Google Scholar
[DT07]Derenthal, U. and Tschinkel, Yu.Universal torsors over del Pezzo surfaces and rational points. In Equidistribution in Number Theory, an Introduction, volume 237 of NATO Sci. Ser. II Math. Phys. Chem. (Springer, Dordrecht, 2007), pages 169196.Google Scholar
[FMT89]Franke, J., Manin, Yu. I. and Tschinkel, Yu.Rational points of bounded height on Fano varieties. Invent. Math. 95 (2) (1989), 421435.Google Scholar
[Pey95]Peyre, E.Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke Math. J. 79 (1) (1995), 101218.Google Scholar
[TT12]Tanimoto, S. and Tschinkel, Yu.Height zeta functions of equivariant compactifications of semi-direct products of algebraic groups. In Zeta Functions in Algebra and Geometry, volume 566 of Contemp. Math. (Amer. Math. Soc., Providence, RI, 2012), pages 119157.CrossRefGoogle Scholar