## On the arithmetic of a family of degree – two $$K3$$ surfaces.(English)Zbl 1410.14029

The paper under review proves the following theorem:
Theorem. Let $${\mathbb{P}}={\mathbb{P}}_{\mathbb{Q}}(1,1,1,3)$$ denote the weighted projective space over $${\mathbb{Q}}$$ with weights $$(1,1,1,3)$$ and coordinates $$x,y,z$$ and $$w$$. Let $${\mathbb{A}}^1$$ be the affine line over $${\mathbb{Q}}$$, with coordinate $$t$$. Consider the family of degree-two $$K3$$ surfaces $$X: w^2=x^6+y^6+x^6+tx^2y^2z^2$$. Let $${\mathfrak{X}}$$ be the generic element of this family, and let $$\overline{\mathfrak{X}}$$ be its base change to $$\overline{\mathbb{Q}(t)}$$.
Then $$\mathfrak{X}$$ is a $$K3$$ surface defined over $${\mathbb{Q}}(t)$$. The geometric Picard lattice $$\text{Pic}\overline{\mathfrak{X}}$$ is isomorphic to the unique (up to isometries) lattice with rank $$19$$, signature $$(1,18)$$, determinant $$2^53^3$$, and discriminant group isomorphic to $${\mathbb{Z}}/6{\mathbb{Z}}\times (\mathbb{Z}/12{\mathbb{Z}})^2$$.
The theorem is proved, first by showing that $${\mathfrak{X}}$$ is isogenous to the Kummer surface associated to the abelian surface $$E\times E$$ where $$E$$ is an elliptic curve with $$j$$-invariant $$-(4t)^3$$. Then the geometric Picard number $$\rho(\overline{\mathfrak{X}})$$ is computed to be $$19$$ and the Picard lattice is explicitly determined. The explicit description of $$\text{Pic}\overline{\mathfrak{X}}$$ enabled them to find the arithmetic invariants.

### MSC:

 14J28 $$K3$$ surfaces and Enriques surfaces

Magma
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### References:

 [1] Bouyer, F., Costa, E., Festi, D., Nicholls, C. and West, M.. Accompanying MAGMA code. http://www.staff.uni-mainz.de/dfesti/AWS2015Magma.txt (2017). [2] Bosma, W., Cannon, J. and Playoust, C.The Magma algebra system. I. The user language. J. Symbolic Comput.24(3-4) (1997), 235-265. Computational algebra and number theory (London, 1993). · Zbl 0898.68039 [3] Barth, W. P., Hulek, K., Peters, C. A. M. and Van De Ven, A. Compact complex surfaces volume 4 of Ergeb. Math. Grenzgeb. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. (Springer-Verlag, Berlin, second edition, 2004). [4] Bogomolov, F. A. and Tschinkel, Yu.Density of rational points on elliptic K3 surfaces. Asian J. Math.4(2) (2000), 351-368. · Zbl 0983.14008 [5] Charles, F., On the Picard number of K3 surfaces over number fields, Algebra Number Theory, 8, 1-17, (2014) · Zbl 1316.14069 [6] Conway, J. H. and Sloane, N. J. A.Sphere packings, lattices and groups. Grundlehren Math Wiss vol. 290 [Fundamental Principles of Mathematical Sciences]. (Springer-Verlag, New York, third edition, 1999). With additional contributions by Bannai, E., Borcherds, R. E., Leech, J., Norton, S. P., Odlyzko, A. M., Parker, R. A., Queen, L. and Venkov, B. B. [7] Dolgachev, I. V.Mirror symmetry for lattice polarised K3 surfaces. J. Math. Sci.81(3) (1996), 2599-2630. Algebraic geometry, 4. · Zbl 0890.14024 [8] Elsenhans, A.-S. and Jahnel, J.K3 surfaces of Picard rank one and degree two. In Algorithmic number theory Lecture Notes in Comput. Sci. vol. 5011 (Springer, Berlin, 2008), pp. 212-225. · Zbl 1205.11073 [9] Elsenhans, A.-S. and Jahnel, J.The Picard group of a K3 surface and its reduction modulo p. Algebra Number Theory5(8) (2011), 1027-1040. · Zbl 1243.14014 [10] Festi, D. Topics in the arithmetic of del Pezzo and K3 surfaces. PhD. thesis. Universiteit Leiden (2016). [11] Hartshorne, R.Algebraic geometry Graduate Texts in Math. No. 52. (Springer-Verlag, New York-Heidelberg, 1977). · Zbl 0367.14001 [12] Hassett, B., Kresch, A. and Tschinkel, Y.Effective computation of Picard groups and Brauer-Manin obstructions of degree two K3 surfaces over number fields. Rendiconti del Circolo Matematico di Palermo62(1) (2013), 137-151. · Zbl 1297.14027 [13] Hindry, M. and Silverman, J. H.Diophantine geometry: an introduction Graduate Texts in Math. vol. 201 (Springer-Verlag, New York, 2000). · Zbl 0948.11023 [14] Huybrechts, D.Lectures on K3 surfaces Camb. Stud. Adv. Math. vol. 158 (Cambridge University Press, Cambridge, 2016). [15] Inose, H. Defining equations of singular K3 surfaces and a notion of isogeny. In Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) (Kinokuniya Book Store, Tokyo, 1978), pp. 495-502. · Zbl 0411.14009 [16] Kresch, A. and Tschinkel, Y.On the arithmetic of del Pezzo surfaces of degree 2. Proc. London Math. Soc. (3) 89(3) (2004), 545-569. · Zbl 1075.14019 [17] Maulik, D. and Poonen, B.Néron-Severi groups under specialisation. Duke Math. J.161(11) (2012), 2167-2206. · Zbl 1248.14011 [18] Mumford, D.Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, No. 5. Published for the Tata Institute of Fundamental Research, Bombay (Oxford University Press, London, 1970). · Zbl 0198.25801 [19] Nikulin, V. V., Integer symmetric bilinear forms and some of their geometric applications, Math USSR-Izv, 14, 103-167, (1980) · Zbl 0427.10014 [20] Nikulin, V. V., Integral symmetric bilinear forms and some of their applications, Mathematics of the USSR-Izvestiya, 14, 103, (1980) · Zbl 0427.10014 [21] Poonen, B., Testa, D. and Luijk, R. VanComputing Néron-Severi groups and cycle class groups. Compositio. Math.151(4) (2015), 713-734. · Zbl 1316.14017 [22] Schütt, M. and Shioda, T.Elliptic surfaces. In Algebraic geometry in East Asia-Seoul 2008 Adv. Stud. Pure Math. vol. 60 (Math. Soc.Japan, Tokyo, 2010), pp. 51-160. · Zbl 1216.14036 [23] Stoll, M. and Testa, D. The surface parametrising cuboids. arXiv:1009.0388 (2010). [24] Várilly-Alvarado, A. and Viray, B.Failure of the Hasse principle for Enriques surfaces. Adv. Math.226(6) (2011), 4884-4901. · Zbl 1239.11068 [25] Van Luijk, R., An elliptic K3 surface associated to Heron triangles, J. Number Theory, 123, 92-119, (2007) · Zbl 1160.14029
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