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Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture

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Abstract

Let \({\mathscr {X}} \rightarrow C\) be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve C in characteristic \(p \ge 5\). We prove that the geometric Picard rank jumps at infinitely many closed points of C. More generally, suppose that we are given the canonical model of a Shimura variety \({\mathcal {S}}\) of orthogonal type, associated to a lattice of signature (b, 2) that is self-dual at p. We prove that any generically ordinary proper curve C in \({\mathcal {S}}_{{\overline{{\mathbb {F}}}}_p}\) intersects special divisors of \({\mathcal {S}}_{{\overline{{\mathbb {F}}}}_p}\) at infinitely many points. As an application, we prove the ordinary Hecke orbit conjecture of Chai–Oort in this setting; that is, we show that ordinary points in \({\mathcal {S}}_{{\overline{{\mathbb {F}}}}_p}\) have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties.

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Notes

  1. Note that the Picard lattice of a K3 surface is equipped with a non-degenerate quadratic form arising from the intersection pairing. The discriminant of the Picard lattice is defined to be the discriminant of this quadratic form.

  2. The special divisors Z(d) and \(Z(m^2d)\) are in the same Hecke orbit.

  3. The conjecture was made in the PEL case, but is expect to hold for Hodge type Shimura varieties too, which includes the case of PEL Shimura varieties.

  4. Being \(\mu \)-ordinary means that this point lies in the open Newton stratum of \({\mathcal {S}}_{{\bar{{\mathbb {F}}}}_p}\) and it means ordinary if the ordinary locus in \({\mathcal {S}}_{{\bar{{\mathbb {F}}}}_p}\) is nonempty, which will be the case for us in the rest of the paper.

  5. Here we only work with the special case when the CM field is an imaginary quadratic field; in the special case when the polarization is principle, see also [29, §2] or [46, §9.3].

  6. Since we work with the hyperspecial case, all the results listed here are in [31] and we follow the convention of using cohomology as in [31].

  7. We drop the ones which do not make sense. For instance, if p is invertible in T, we drop \({\mathrm {cris}}\); if \(T_{\mathbb {Q}}=\emptyset \), we drop B.

  8. By [24, Lem. 4.2.4], this definition of being supersingular is equivalent to that the corresponding Kuga–Satake abelian variety is supersingular.

  9. We note that our notation differs from Ogus’ by a Tate twist, so our \(F^1_{\text {con}}\) corresponds to Ogus’ \(F^2_{\text {con}}\)

  10. By [39, p. 327], \(t_P/2\) is the Artin invariant if \(A_P\) is the Kuga–Satake abelian variety associated to a K3 surface.

  11. Ogus proved that the isomorphism classes of so-called K3 crystals ([37, Def. 3.1]) are in bijection with the data in [37, Thm. 3.5] described here; indeed, the isomorphism classes of K3 crystals in Ogus’s sense are isomorphism classes of \({\mathbb {L}}\) for supersingular points by [24].

  12. All empty entries in the matrix are 0.

  13. Comparing to [33, §3], §3.2.1 in loc. cit. is a special case of the split even dimensional case, §3.2.2 in loc. cit. is a special case of the non-split even dimensional case, and §3.3 in loc. cit. is a special case of the odd dimensional case.

  14. We would like to thank the anonymous referee for pointing out this reference to us.

  15. For more details, see [33, Proof of Thm. 5.1.2 assuming Prop. 5.1.3].

  16. Here we have \(p^{r+2}\) instead of \(p^{r+1}\) is due to the fact that \({{\,\mathrm{Span}\,}}_W\{e_i,f_i, e'_j, f'_j\}\ne {\mathbb {L}}\) but we still have \({{\,\mathrm{Span}\,}}_W\{e_i,f_i, e'_j, f'_j\}\supset p {\mathbb {L}}\).

  17. Note that we also consider B(x) and \(K_i\) as the zeroth \(\sigma \)-twist of themselves.

  18. Here and also in the statement of Theorem 5.15, by the Newton stratum associated to \(\nu \) in Kottwitz’s set, we mean the closed subscheme in the Shimura variety parametrizing points whose Newton points/polygons \(\nu '\le \nu \) with respect to the partial order in Kottwitz’s set; equivalently, this closed subscheme is the Zariski closure of the locally closed subscheme parametrizing points whose Newton points/polygons are exactly \(\nu \). We refer to this locally closed subscheme as the open Newton stratum.

  19. There is another possible choice with \(\lambda \) replaced by \(-\lambda \); given the computation is the same for both cases, we will just work with the first case.

  20. Indeed, by [46, Lem. 4.7], every \(m\gg 1\) is representable since L is maximal.

  21. Here we follow the convention in [32, §3.1] that we use an embedding into the group of symplectic similitudes of a symplectic space over \({\mathbb {Q}}\) which admits a self-dual \({\mathbb {Z}}\)-lattice; this embedding may be different from the one in Sect. 2.2, but can be constructed from the one in Sect. 2.2 using Zarhin’s trick as explained in [32, p. 442].

  22. In [32] there is a twist by \(2\pi i\) which we are suppressing.

  23. Note that by definition in [32, §2.1.11], \({\mathbf {B}}_{K_n}\otimes {\mathbb {Z}}[1/\ell ]\) is independent of n for our \(K_n\).

  24. Here we call \({\mathbf {H}}^*\) in [32, §2.1.22] the rational closure of the cone \({\mathbf {H}}\).

  25. This means that \([-,-]\) induces an isomorphism between \({{\,\mathrm{Span}\,}}_{{\mathbb {Z}}}\{\zeta ,\omega \}\) and \({{\,\mathrm{Hom}\,}}(I,{\mathbb {Z}})\) with \(\zeta ,\omega \) mapping to the basis dual to \(\{z,w\}\); the existence of such a basis is given by [51, Def. 2.1, Lem. 2.2].

  26. because \({\mathcal {S}}_{{\mathbb {F}}_p}^{\mathrm {BB}}\) is projective by [32, Thm. 3]

  27. More precisely, as explained on [2, p. 434] that \({\mathcal {Z}}(m)_{{\mathbb {Q}}}\) is a finite disjoint union of GSpin Shimura varieties associated to quadratic spaces isomorphic to \((v^\perp , Q|_{v^\perp })\subset (V,Q)\), where \(v\in L\) with \(Q(v)=m\); since \(p\not \mid m\), then \((v^\perp , Q|_{v^\perp })\) is self-dual at p and the embedding \(v^\perp \subset V\) also induces a hyperspecial level for the Shimura variety associated to \(v^\perp \). By [2, Prop. 4.4.2] (or [31, Cor. 6.23]), \({\mathcal {Z}}(m)\) is normal and flat and thus is the disjoint union of canonical integral models of the GSpin Shimura varieties associated to isomorphism classes of \(v^\perp \) with \(Q(v)=m\).

  28. Here we work with isomorphism classes of abelian varieties; one may also describe the moduli problem in the prime-to-p isogeny category of abelian varieties; see for instance [30, §11.2] for a short summary of the discussion in [41, 42].

  29. Even though [7] works with principal polarization case, since we work with hyperspecial level at p, the description also applies to our case here.

References

  1. André, Y.: Pour une théorie inconditionnelle des motifs. Inst. Hautes Études Sci. Publ. Math. 83, 5–49 (1996). (French)

    Article  Google Scholar 

  2. Andreatta, F., Goren, E.Z., Howard, B., Madapusi Pera, K.: Faltings heights of abelian varieties with complex multiplication. Ann. Math. (2) 187(2), 391–531 (2018)

    Article  MathSciNet  Google Scholar 

  3. Ash, A., Mumford, D., Rapoport, M., Tai, Y.-S.: Smooth Compactifications of Locally Symmetric Varieties, 2nd edn. Cambridge Mathematical Library, Cambridge University Press, Cambridge (2010). (with the collaboration of Peter Scholze)

  4. Bary-Soroker, L., Entin, A.: Explicit Hilbert’s Irreducibility Theorem in Function Fields, Abelian Varieties and Number Theory, Contemp. Math., vol. 767, pp. 125–134. Amer Math Soc, Providence (2021)

    Google Scholar 

  5. Borcherds, R.E.: The Gross–Kohnen–Zagier theorem in higher dimensions. Duke Math. J. 97(2), 219–233 (1999)

    Article  MathSciNet  Google Scholar 

  6. Bruinier, J.H.: Borcherds products with prescribed divisor. Bull. Lond. Math. Soc. 49(6), 979–987 (2017)

    Article  MathSciNet  Google Scholar 

  7. Bruinier, J.H., Howard, B., Kudla, S.S., Rapoport, M., Yang, T.: Modularity of generating series of divisors on unitary Shimura varieties, Astérisque 421, Diviseurs arithmétiques sur les variétés orthogonales et unitaires de Shimura, pp. 7-125 (2020)

  8. Bruinier, J.H., Kuss, M.: Eisenstein series attached to lattices and modular forms on orthogonal groups. Manuscr. Math. 106(4), 443–459 (2001)

    Article  MathSciNet  Google Scholar 

  9. Bruinier, J.H., Zemel, S.: Special cycles on toroidal compactifications of orthogonal Shimura varieties. Math. Ann. (2021) (published online)

  10. Chai, C.-L.: Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli. Invent. Math. 121(3), 439–479 (1995)

    Article  MathSciNet  Google Scholar 

  11. Chai, C.-L.: Families of ordinary abelian varieties: canonical coordinates, p-adic monodromy, Tate-linear subvarieties and Hecke orbits (2003). https://www.math.upenn.edu/chai

  12. Chai, C.-L.: Hecke orbits on Siegel modular varieties, Geometric methods in algebra and number theory. Progr. Math., vol. 235, pp. 71–107. Birkhäuser Boston, Boston (2005)

  13. Chai, C.-L.: Hecke orbits as Shimura varieties in positive characteristic. International Congress of Mathematicians, vol. II, pp. 295–312. Eur. Math. Soc, Zürich (2006)

  14. Chai, C.-L., Oort, F.: Hypersymmetric abelian varieties. Pure Appl. Math. Q. 2, no. 1. Special Issue: In honor of John H. Coates., pp. 1–27 (2006)

  15. Chai, C.-L., Oort, F.: Moduli of abelian varieties and p-divisible groups, Arithmetic geometry. Clay Math. Proc., vol. 8, pp. 441–536. Amer. Math. Soc., Providence (2009)

  16. Chai, C.-L., Oort, F.: The Hecke orbit conjecture: a survey and outlook, Open problems in arithmetic algebraic geometry, Adv. Lect. Math. (ALM), vol. 46, pp. 235–262. Int. Press, Somerville (2019)

  17. Charles, F.: Exceptional isogenies between reductions of pairs of elliptic curves. Duke Math. J. 167(11), 2039–2072 (2018)

    Article  MathSciNet  Google Scholar 

  18. Clozel, L., Hee, O., Ullmo, E.: Hecke operators and equidistribution of Hecke points. Invent. Math. 144(2), 327–351 (2001)

    Article  MathSciNet  Google Scholar 

  19. de Jong, A.J.: Crystalline Dieudonné module theory via formal and rigid geometry. Inst. Hautes Études Sci. Publ. Math. 82(1995), 5–96 (1996)

    MATH  Google Scholar 

  20. Eskin, A., Katznelson, Y.R.: Singular symmetric matrices. Duke Math. J. 79(2), 515–547 (1995). https://doi.org/10.1215/S0012-7094-95-07913-7

    Article  MathSciNet  MATH  Google Scholar 

  21. Hanke, J.: Local densities and explicit bounds for representability by a quadratric form. Duke Math. J. 124(2), 351–388 (2004)

    Article  MathSciNet  Google Scholar 

  22. Howard, B.: Complex multiplication cycles and Kudla–Rapoport divisors. II. Am. J. Math. 137(3), 639–698 (2015)

    Article  MathSciNet  Google Scholar 

  23. Howard, B., Pera, M.K.: Arithmetic of Borcherds products, Astérisque 421. Diviseurs arithmétiques sur les variétés orthogonales et unitaires de Shimura 187–297 (2020)

  24. Howard, B., Pappas, G.: Rapoport–Zink spaces for spinor groups. Compos. Math. 153(5), 1050–1118 (2017)

    Article  MathSciNet  Google Scholar 

  25. Iwaniec, H.: Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, vol. 17. American Mathematical Society, Providence (1997)

    MATH  Google Scholar 

  26. Keating, K.: Lifting endomorphisms of formal A-modules. Compos. Math. 67(2), 211–239 (1988)

    MathSciNet  MATH  Google Scholar 

  27. Kisin, M.: Integral models for Shimura varieties of abelian type. J. Am. Math. Soc. 23(4), 967–1012 (2010)

    Article  MathSciNet  Google Scholar 

  28. Kisin, M.: mod p points on Shimura varieties of abelian type. J. Am. Math. Soc. 30(3), 819–914 (2017)

    Article  MathSciNet  Google Scholar 

  29. Kudla, S., Rapoport, M.: Special cycles on unitary Shimura varieties II: global theory. J. Reine Angew. Math. 697, 91–157 (2014)

    MathSciNet  MATH  Google Scholar 

  30. Li, C., Zhang, W.: Kudla–Rapoport cycles and derivatives of local densities. J. Am. Math. Soc. (2021) (published online)

  31. Madapusi, K.: Integral canonical models for spin Shimura varieties. Compos. Math. 152(4), 769–824 (2016)

    Article  MathSciNet  Google Scholar 

  32. Madapusi Pera, K.: Toroidal compactifications of integral models of Shimura varieties of Hodge type. Ann. Sci. Éc. Norm. Supér. (2) 52(2), 393–514 (2019). (English, with English and French summaries)

    Article  MathSciNet  Google Scholar 

  33. Maulik, D., Shankar, A.N., Tang, Y.: Reductions of abelian surfaces over global function fields. Compos. Math. (to appear)

  34. Moonen, B.: Models of Shimura varieties in mixed characteristics. Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Math. Soc. Lecture Note Ser., vol. 254, pp. 267–350. Cambridge Univ. Press, Cambridge (1998)

  35. Noot, R.: Models of Shimura varieties in mixed characteristic. J. Algebraic Geom. 5(1), 187–207 (1996)

    MathSciNet  MATH  Google Scholar 

  36. Oguiso, K.: Local families of K3 surfaces and applications. J. Algebraic Geom. 12(3), 405–433 (2003)

    Article  MathSciNet  Google Scholar 

  37. Ogus, A.: Supersingular K3 crystals, Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Astérisque, vol. 64, pp. 3–86. Soc. Math. France, Paris (1979)

  38. Ogus, A.: Hodge Cycles and Crystalline Cohomology, vol. 900, pp. 357–414. Springer, Berlin (1982)

    MATH  Google Scholar 

  39. Ogus, A.: Singularities of the height strata in the moduli of K3 surfaces. Moduli of Abelian Varieties (Texel Island, 1999). Progr. Math., vol. 195. Birkhäuser, Basel, pp. 325–343 (2001)

  40. Pink, R.: Arithmetical compactification of mixed Shimura varieties, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 209, Universität Bonn, Mathematisches Institut, Bonn. Dissertation, p. 1989. Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn (1990)

  41. Rapoport, M., Smithling, B., Zhang, W.: Arithmetic diagonal cycles on unitary Shimura varieties. Compos. Math. 156(9), 1745–1824 (2020)

    Article  MathSciNet  Google Scholar 

  42. Rapoport, M., Smithling, B., Zhang, W.: On Shimura varieties for unitary groups. Pure Appl. Math. Q. 17(2), 773–837 (2021)

    Article  MathSciNet  Google Scholar 

  43. Sarnak, P.: Some Applications of Modular Forms, Cambridge Tracts in Mathematics, vol. 99. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  44. Serre, J.-P.: Lectures on the Mordell-Weil theorem, Aspects of Mathematics, E15, Friedr (translated from the French and edited by Martin Brown from notes by Michel Waldschmidt). Vieweg & Sohn, Braunschweig (1989)

    Google Scholar 

  45. Shankar, A.N.: The Hecke orbit conjecture for “modèles étranges” (preprint)

  46. Shankar, A.N., Shankar, A., Tang, Y., Tayou, S.: Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields. arXiv: 1909.07473

  47. Shankar, A.N., Tang, Y.: Exceptional splitting of reductions of abelian surfaces. Duke Math. J. 169(3), 397–434 (2020)

    Article  MathSciNet  Google Scholar 

  48. Tayou, S.: On the equidistribution of some Hodge loci. J. Reine Angew. Math. 762, 167–194 (2020)

    Article  MathSciNet  Google Scholar 

  49. Voisin, C.: Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés [Specialized Courses], vol. 10, p. viii+595. Société Mathématique de France, Paris (2002) (French)

  50. Xiao, L.X.: On the Hecke orbit conjecture for PEL type Shimura varieties. arXiv:2006.06859

  51. Zemel, S.: The structure of integral parabolic subgroups of orthogonal groups. J. Algebra 559, 95–128 (2020)

    Article  MathSciNet  Google Scholar 

  52. Zhou, R.: Motivic cohomology of quaternionic Shimura varieties and level raising. arXiv:1901.01954

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Acknowledgements

We thank George Boxer, Ching-Li Chai, Johan de Jong, Kai-Wen Lan, Keerthi Madapusi Pera, Frans Oort, Arul Shankar, Andrew Snowden, Salim Tayou, and Tonghai Yang for helpful discussions, as well as Arthur and D.W. Read for additional assistance. A.S. has been partially supported by the NSF Grant DMS-2100436 and Y.T. has been partially supported by the NSF Grant DMS-1801237. Y.T. was a chargée de recherche at CNRS and Université Paris-Saclay from February 2020 to June 2021. We would like to thank the referee for a careful and thorough reading, and for valuable suggestions which vastly improved the exposition of the paper.

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  1. Department of mathematics, Massachusetts Institute of Technology, Cambridge, USA

    Davesh Maulik

  2. Department of mathematics, University of Wisconsin, Madison, USA

    Ananth N. Shankar

  3. Department of mathematics, Princeton University, Princeton, USA

    Yunqing Tang

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  1. Davesh Maulik
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Maulik, D., Shankar, A.N. & Tang, Y. Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture. Invent. math. 228, 1075–1143 (2022). https://doi.org/10.1007/s00222-022-01097-x

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