On the distribution of the Picard ranks of the reductions of a \(K3\) surface. (English) Zbl 1471.14080

Let \(S/K\) be a \(K3\) surface over a number field \(K\) and \(\overline{K}\) a fixed algebraic closure of \(K\). Let \(\mathcal{O}_K\) denote the ring of integers of \(K\), \(\mathfrak{p}\in \mathcal{O}_K\) a prime, \(\mathbb{F}_{\mathfrak{p}}\) the residue field of \(\mathfrak{p}\), and \(\overline{\mathbb{F}}_\mathfrak{p}\) a fixed algebraic closure of \(\mathbb{F}_{\mathfrak{p}}\). Assume that \(\mathfrak{p}\) is a prime of good reduction for \(S\) and let \(S_{\mathbb{F}_\mathfrak{p}}/\mathbb{F}_\mathfrak{p}\) denote the reduction of \(S\) modulo \(\mathfrak{p}\). It is well known that the geometric Picard number of \(S_{\mathbb{F}_\mathfrak{p}}\), denoted by \(\rho (S_{\overline{\mathbb{F}}_\mathfrak{p}}) := \mathrm{rk}\, \mathrm{Pic} ( S_{\overline{\mathbb{F}}_\mathfrak{p}})\), is always greater than the geometric Picard number of \(S_{\overline{K}}\). In other words, we always have the following inequality: \[ \rho (S_{\overline{\mathbb{F}}_\mathfrak{p}}) \geq \rho (S_{\overline{K}}). \]
In the paper under review, the authors investigate the primes of good reduction for \(S\) for which the inequality above is strict, called jump primes. We know of two cases for which every prime of good reduction is a jump prime:
when \(\rho (S_{\overline{K}})\) is odd, by the (now proven) Tate conjecture, and
when \(S\) has real multiplication by an endomorphism field \(E\) and the integer \((22-\rho (S_{\overline{K}}))/[E:\mathbb{Q}]\) is odd, by work of [F. Charles, Algebra Number Theory 8, No. 1, 1–17 (2014; Zbl 1316.14069)].
The same work by F. Charles also shows that these are the only cases in which all primes of good reduction are jump primes.
The authors of the paper under review introduce the quantities \(\Delta_{H^2}(S)\) and \(\Delta_{\mathrm{Pic}}(S)\), together with the quadratic character \[ \tau_S\colon \mathfrak{p}\mapsto \left( \frac{\Delta_{H^2}(S)\Delta_{\mathrm{Pic}}(S)}{\mathfrak{p}}\right), \] called the transcendental character of the \(K3\) surface \(S\). The main result of the paper shows that if \(S\) has even geometric Picard number and \(\tau_S (\mathfrak{p})=-1\), then \[ \rho (S_{\overline{\mathbb{F}}_\mathfrak{p}}) \geq \rho (S_{\overline{K}})+2\; . \] Using this, the authors prove that the experimental observation by [E. Costa and Y. Tschinkel, Exp. Math. 23, No. 4, 475–481 (2014; Zbl 1311.14039)] on the density of jump primes is correct. The authors also present an algorithm to compute \(\tau_S\) for a given \(K3\) surface embedded in a projective space, alongside with explicit examples. Finally, as an application of the main result, they show that if \(\rho (S_{\overline{K}})\) is even, \(S_{\overline{K}}\) has neither real nor complex multiplication, and the product \(\Delta_{H^2}(S)\Delta_{\mathrm{Pic}}(S)\) is not a square in \(K\), then \(S_{\overline{K}}\) contains infinitely many rational curves.
Reviewer: Dino Festi (Milan)


14J28 \(K3\) surfaces and Enriques surfaces
14F20 Étale and other Grothendieck topologies and (co)homologies
11G35 Varieties over global fields
14G25 Global ground fields in algebraic geometry


SageMath; Magma; NTL; FLINT
Full Text: DOI arXiv


[1] André, Y., Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses 17 (2004), Paris: Société Mathématique de France, Paris · Zbl 1060.14001
[2] Barth, W.; Hulek, K.; Peters, C.; van de Ven, A., Compact Complex Surfaces, Second edition (2004), Berlin: Springer, Berlin · Zbl 1036.14016
[3] Benoist, O., Construction de courbes sur les surfaces K3 (d’aprés Bogomolov-Hassett-Tschinkel, Charles, Li-Liedtke, Madapusi Pera, Maulik \(\ldots )\), Astérisque, 367-368, 219-253 (2015) · Zbl 1356.14001
[4] Berndt, BC; Evans, RJ, Sums of Gauss, Jacobi, and Jacobsthal, J. Number Theory, 11, 349-398 (1979) · Zbl 0412.10027
[5] Bogomolov, FA; Hassett, B.; Tschinkel, Yu, Constructing rational curves on \(K3\) surfaces, Duke Math. J., 157, 535-550 (2011) · Zbl 1236.14035
[6] Bogomolov, FA; Tschinkel, Yu, Density of rational points on elliptic \(K3\) surfaces, Asian J. Math., 4, 351-368 (2000) · Zbl 0983.14008
[7] Bosma, W.; Cannon, J.; Playoust, C., The Magma algebra system I, The user language, J. Symb. Comput., 24, 235-265 (1997) · Zbl 0898.68039
[8] Bouyer, F.; Costa, E.; Festi, D.; Nicholls, C.; West, M., On the arithmetic of a family of degree-two \(K3\) surfaces, Math. Proc. Camb. Philos. Soc., 166, 523-542 (2019) · Zbl 1410.14029
[9] Bright, M.J.: Computations on diagonal quartic surfaces, Ph.D. thesis, Cambridge (2002)
[10] Charles, F., The Tate conjecture for \(K3\) surfaces over finite fields, Invent. Math., 194, 119-145 (2013) · Zbl 1282.14014
[11] Charles, F., On the Picard number of \(K3\) surfaces over number fields, Algebra Number Theory, 8, 1-17 (2014) · Zbl 1316.14069
[12] Chen, X., Gounelas, F., Liedtke, C.: Curves on \(K3\) surfaces. arXiv:1907.01207
[13] Costa, E.; Tschinkel, Yu, Variation of Néron-Severi ranks of reductions of \(K3\) surfaces, Exp. Math., 23, 475-481 (2014) · Zbl 1311.14039
[14] Deligne, P., La conjecture de Weil I, Publ. Math. IHES, 43, 273-307 (1974) · Zbl 0287.14001
[15] Deligne, P., La conjecture de Weil II, Publ. Math. IHES, 52, 137-252 (1980) · Zbl 0456.14014
[16] Elsenhans, A.-S., Jahnel, J.: On Weil polynomials of \(K3\) surfaces. In: Algorithmic Number Theory (ANTS 9). Lecture Notes in Computer Science, vol. 6197, pp. 126-141. Springer, Berlin (2010) · Zbl 1260.11046
[17] Elsenhans, A-S; Jahnel, J., On the computation of the Picard group for \(K3\) surfaces, Math. Proc. Camb. Philos. Soc., 151, 263-270 (2011) · Zbl 1223.14044
[18] Elsenhans, A-S; Jahnel, J., Kummer surfaces and the computation of the Picard group, LMS J. Comput. Math., 15, 84-100 (2012) · Zbl 1297.14043
[19] Elsenhans, A-S; Jahnel, J., Examples of \(K3\) surfaces with real multiplication, in: Proceedings of the ANTS XI conference (Gyeongju 2014), LMS J. Comput. Math., 17, 14-35 (2014) · Zbl 1307.14063
[20] Elsenhans, A-S; Jahnel, J., On the characteristic polynomial of the Frobenius on étale cohomology, Duke Math. J., 164, 2161-2184 (2015) · Zbl 1348.14056
[21] Elsenhans, A-S; Jahnel, J., Point counting on \(K3\) surfaces and an application concerning real and complex multiplication, in: Proceedings of the ANTS XII conference (Kaiserslautern 2016), LMS J. Comput. Math., 19, 12-28 (2016) · Zbl 1361.14026
[22] Fontaine, J-M, Représentations \(l\)-adiques potentiellement semi-stables, Astérisque, 223, 321-347 (1994) · Zbl 0873.14020
[23] Hart, W., Johansson, F., Pancratz, S.: FLINT: Fast Library for Number Theory. http://flintlib.org
[24] Huybrechts, D., Lectures on K3 Surfaces (2016), Cambridge: Cambridge University Press, Cambridge
[25] Ireland, KF; Rosen, MI, A Classical Introduction to Modern Number Theory (1982), New York: Springer, New York
[26] Kim, W.; Madapusi Pera, K., 2-adic integral canonical models, Forum Math. Sigma, 4, e28 (2016) · Zbl 1362.11059
[27] Li, J.; Liedtke, Ch, Rational curves on \(K3\) surfaces, Invent. Math., 188, 713-727 (2012) · Zbl 1255.14026
[28] Lieblich, M.; Maulik, D.; Snowden, A., Finiteness of \(K3\) surfaces and the Tate conjecture, Ann. Sci. École Norm. Sup., 47, 285-308 (2014) · Zbl 1329.14078
[29] Livné, R., Motivic orthogonal two-dimensional representations of \({\text{Gal}}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})\), Israel J. Math., 92, 149-156 (1995) · Zbl 0847.11035
[30] Madapusi Pera, K., The Tate conjecture for \(K3\) surfaces in odd characteristic, Invent. Math., 201, 625-668 (2015) · Zbl 1329.14079
[31] Matsusaka, T., The criteria for algebraic equivalence and the torsion group, Am. J. Math., 79, 53-66 (1957) · Zbl 0077.34303
[32] Neukirch, J., Algebraic Number Theory (1999), Berlin: Springer, Berlin
[33] Ochiai, T., \(l\)-independence of the trace of monodromy, Math. Ann., 315, 321-340 (1999) · Zbl 0980.14014
[34] Pinch, R.G.E., Swinnerton-Dyer, H.P.F.: Arithmetic of diagonal quartic surfaces I. In: \(L\)-functions and arithmetic (Durham 1989). London Math. Soc. Lecture Note Ser., vol. 153, pp. 317-338. Cambridge Univ. Press, Cambridge (1991) · Zbl 0736.14006
[35] Saito, T., The discriminant and the determinant of a hypersurface of even dimension, Math. Res. Lett., 19, 855-871 (2012) · Zbl 1285.14046
[36] Serre, J.-P.: Facteurs locaux des fonctions zêta des varietés algébriques (définitions et conjectures). In: Séminaire Delange-Pisot-Poitou (Théorie des nombres), 11e année 1969/70, Exposé 19, pp. 1-19. Secrétariat Math., Paris (1970)
[37] Artin, M., Grothendieck, A. et Verdier, J.-L. (avec la collaboration de Deligne, P. et Saint-Donat, B.): Théorie des topos et cohomologie étale des schémas, Séminaire de Géométrie Algébrique du Bois Marie 1963-1964 (SGA 4). In: Lecture Notes in Math. vol. 269, 270, 305. Springer, Berlin, Heidelberg, New York (1972-1973)
[38] Deligne, P. et Katz, N.: Groupes de Monodromie en Géométrie Algébrique, Séminaire de Géométrie Algébrique du Bois Marie 1967-1969 (SGA 7). In: Lecture Notes in Math. vol. 288, 340, Springer, Berlin, Heidelberg, New York (1973)
[39] Shioda, T., Inose, H.: On singular \(K3\) surfaces. In: Complex Analysis and Algebraic Geometry, pp. 119-136. Iwanami Shoten, Tokyo (1977) · Zbl 0374.14006
[40] Shoup, V.: NTL: Number Theory Library. http://www.shoup.net/ntl/
[41] Spanier, EH, Algebraic Topology (1966), New York: McGraw-Hill Book Co., New York
[42] Stein, W.A. et al.: Sage Mathematics Software (Version 7.3). The Sage Development Team (2016) http://www.sagemath.org
[43] Suh, J., Symmetry and parity in Frobenius action on cohomology, Compos. Math., 148, 295-303 (2012) · Zbl 1258.14023
[44] Tankeev, SG, Surfaces of \(K3\) type over number fields and the Mumford-Tate conjecture (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 54, 846-861 (1990)
[45] Tankeev, SG, Surfaces of \(K3\) type over number fields and the Mumford-Tate conjecture II (Russian), Izv. Ross. Akad. Nauk Ser. Mat., 59, 179-206 (1995)
[46] Tate, J.T.: Algebraic cycles and poles of zeta functions. In: Arithmetical Algebraic Geometry. Proc. Conf. Purdue Univ. 1963, Harper & Row, New York, pp. 93-110 (1965)
[47] van Luijk, R., \(K3\) surfaces with Picard number one and infinitely many rational points, Algebra Number Theory, 1, 1-15 (2007) · Zbl 1123.14022
[48] Wall, CTC, Singular Points of Plane Curves (2004), Cambridge: Cambridge University Press, Cambridge
[49] Warner, FW, Foundations of Differentiable Manifolds and Lie groups, Scott (1971), Glenview-London: Foresman and Co., Glenview-London
[50] Zarhin, YuG, Hodge groups of \(K3\) surfaces, J. Reine Angew. Math., 341, 193-220 (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.