We study complex algebraic K3 surfaces with finite automorphism groups and polarized by rank 14, 2-elementary lattices. Three such lattices exist, namely, $H \oplus E_8(-1) \oplus A_1(-1)^{\oplus 4}$ , $H \oplus E_8(-1) \oplus D_4(-1)$ , and $H \oplus D_8(-1) \oplus D_4(-1)$ . As part of our study, we provide birational models for these surfaces as quartic projective hypersurfaces and describe the associated coarse moduli spaces in terms of suitable modular invariants. Additionally, we explore the connection between these families and dual K3 families related via the Nikulin construction.

REFERENCES

1 F. Balestrieri, J. Desjardins, A. Garbagnati, C. Maistret, C. Salgado, and I. Vogt, Elliptic fibrations on covers of the elliptic modular surface of level 5, I. I. Bouw, E. Ozman, J. Johnson-Leung and R. Newton (Eds.), Women in Numbers Europe II, Assoc. Women Math. Ser., vol. 11, Springer, Cham, 2018, pp. 159–197, DOI https://doi.org/10.1007/978-3-319-74998-3_9. https://mathscinet-ams-org.ezproxy.lib.uconn.edu/mathscinet-getitem?mr=3882710.
10.1007/978-3-319-74998-3_9
Google Scholar
2S.-M. Belcastro, Picard lattices of families of $K3$ surfaces, Comm. Algebra 30 (2002), no. 1, 61–82. https://mathscinet.ams.org/mathscinet-getitem?mr=1880661.
10.1081/AGB-120006479
Web of Science® Google Scholar
3N. Braeger, A. Malmendier, and Y. Sung, Kummer sandwiches and Greene-Plesser construction, J. Geom. Phys. 154 (2020), 103718. https://mathscinet.ams.org/mathscinet-getitem?mr=4099481.
10.1016/j.geomphys.2020.103718
Web of Science® Google Scholar
4A. P. Braun, Y. Kimura, and T. Watari, On the classification of elliptic fibrations modulo isomorphism on k3 surfaces with large Picard number, 2013. https://arxiv.org/pdf/1312.4421.pdf.
Google Scholar
5A. Clingher, R. Donagi, and M. Wijnholt, The Sen limit, Adv. Theor. Math. Phys. 18 (2014), no. 3, 613–658. https://mathscinet.ams.org/mathscinet-getitem?mr=3274790.
10.4310/ATMP.2014.v18.n3.a2
Web of Science® Google Scholar
6A. Clingher and C. F. Doran, Modular invariants for lattice polarized $K3$ surfaces, Michigan Math. J. 55 (2007), no. 2, 355–393, DOI http://doi.org/10.1307/mmj/1187646999.
10.1307/mmj/1187646999
Google Scholar
7A. Clingher and C. F. Doran, On $K3$ surfaces with large complex structure, Adv. Math. 215 (2007), no. 2, 504–539. https://mathscinet.ams.org/mathscinet-getitem?mr=2355598.
10.1016/j.aim.2007.04.014
Web of Science® Google Scholar
8A. Clingher and C. F. Doran, Note on a geometric isogeny of K3 surfaces, Int. Math. Res. Not. 2011 (2011), no. 16, 3657–3687.
10.1093/imrn/rnq230
Web of Science® Google Scholar
9A. Clingher and C. F. Doran, Lattice polarized K3 surfaces and Siegel modular forms, Adv. Math. 231 (2012), no. 1, 172–212, DOI http://doi.org/10.1016/j.aim.2012.05.001.
10.1016/j.aim.2012.05.001
Web of Science® Google Scholar
10A. Clingher, C. F. Doran, and A. Malmendier, Special function identities from superelliptic Kummer varieties, Asian J. Math. 21 (2017), no. 5, 909–951. https://mathscinet.ams.org/mathscinet-getitem?mr=3767270.
10.4310/AJM.2017.v21.n5.a6
Web of Science® Google Scholar
11A. Clingher, T. Hill, and A. Malmendier, Jacobian elliptic fibrations on a special family of K3 surfaces of Picard rank sixteen, Albanian J. Math., 17 (2019), no. 1, 13–28.
Google Scholar
12A. Clingher, T. Hill, and A. Malmendier, The duality between F-theory and the heterotic string in $D=8$ with two Wilson lines, Lett. Math. Phys. 110 (2020), no. 11, 3081–3104. https://mathscinet.ams.org/mathscinet-getitem?mr=4160930.
10.1007/s11005-020-01323-8
Web of Science® Google Scholar
13A. Clingher and A. Malmendier, Nikulin involutions and the CHL string, Comm. Math. Phys. 370 (2019), no. 3, 959–994. https://mathscinet.ams.org/mathscinet-getitem?mr=3995925.
10.1007/s00220-019-03296-9
Web of Science® Google Scholar
14A. Clingher and A. Malmendier, Normal forms for Kummer surfaces, R. Donagie and T. Shaska (Eds.), Integrable Systems and Algebraic Geometry (London Mathematical Society Lecture Note Series), London Mathematical Society Lecture Note Series, pp. Vii-Viii, vol. 2, Cambridge University Press, Cambridge, 2020, pp. 119–174, DOI https://doi.org/10.1017/9781108773355.
10.1017/9781108773355.006
Google Scholar
15A. Clingher, A. Malmendier, and T. Shaska, Six line configurations and string dualities, Comm. Math. Phys. 371 (2019), no. 1, 159–196. https://mathscinet.ams.org/mathscinet-getitem?mr=4015343.
10.1007/s00220-019-03372-0
Web of Science® Google Scholar
16A. Clingher, A. Malmendier, and T. Shaska, On isogenies among certain abelian surfaces, Michigan Math. J. 71 (2022), no. 2, 227–269.
10.1307/mmj/20195790
Web of Science® Google Scholar
17P. Comparin and A. Garbagnati, Van Geemen-Sarti involutions and elliptic fibrations on $K3$ surfaces double cover of $\mathbb {P}^2$ , J. Math. Soc. Japan 66 (2014), no. 2, 479–522. https://mathscinet-ams-org.ezproxy.lib.uconn.edu/mathscinet-getitem?mr=3201823.
10.2969/jmsj/06620479
Web of Science® Google Scholar
18I. Dolgachev, Integral quadratic forms: applications to algebraic geometry (after V. Nikulin), Bourbaki seminar, Vol. 1982/83, Astérisque, 105, (1983) 251–278. https://mathscinet.ams.org/mathscinet-getitem?mr=728992.
Google Scholar
19I. V. Dolgachev, Mirror symmetry for lattice polarized $K3$ surfaces, J. Math. Sci. 81 (1996), no. 3, 2599–2630, DOI http://doi.org/10.1007/BF02362332, algebraic geometry, 4.
10.1007/BF02362332
Google Scholar
20D. Festi and D. C. Veniani, Counting ellptic fibrations on K3 surfaces, 2020, arXiv:2102.09411.
Google Scholar
21A. Garbagnati and C. Salgado, Elliptic fibrations on K3 surfaces with a non-symplectic involution fixing rational curves and a curve of positive genus, Rev. Mat. Iberoam. 36 (2020), no. 4, 1167–1206. https://mathscinet-ams-org.ezproxy.lib.uconn.edu/mathscinet-getitem?mr=4130832.
10.4171/rmi/1163
Web of Science® Google Scholar
22A. Garbagnati and A. Sarti, On symplectic and non-symplectic automorphisms of K3 surfaces, Rev. Mat. Iberoam. 29 (2013), no. 1, 135–162. https://mathscinet.ams.org/mathscinet-getitem?mr=3010125.
10.4171/rmi/716
Web of Science® Google Scholar
23A. Garbagnati and A. Sarti, Kummer surfaces and K3 surfaces with $(\mathbb {Z}/2\mathbb {Z})^4$ symplectic action, Rocky Mountain J. Math. 46 (2016), no. 4, 1141–1205. https://mathscinet.ams.org/mathscinet-getitem?mr=3563178.
10.1216/RMJ-2016-46-4-1141
Web of Science® Google Scholar
24E. Griffin and A. Malmendier, Jacobian elliptic Kummer surfaces and special function identities, Commun. Number Theory Phys. 12 (2018), no. 1, 97–125. https://mathscinet.ams.org/mathscinet-getitem?mr=3798883.
10.4310/CNTP.2018.v12.n1.a4
Web of Science® Google Scholar
25J. Gu and H. Jockers, Nongeometric $F$ -theory–heterotic duality, Phys. Rev. D 91 (2015), no. 8, 086007, 10. https://mathscinet.ams.org/mathscinet-getitem?mr=3417046.
Web of Science® Google Scholar
26W. L. Hoyt, Notes on elliptic $K3$ surfaces, D. V. Chudnovsky, G. V. Chudnovsky, H. Cohn and M. B. Nathanson (Eds.), Number theory (New York, 1984–1985), Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 196–213, DOI https://doi.org/10.1007/BFb0072981.
10.1007/BFb0072981
Google Scholar
27W. L. Hoyt, Elliptic fiberings of Kummer surfaces, D. V. Chudnovsky, G. V. Chudnovsky, H. Cohn and M. B. Nathanson (Eds.), Number theory (New York, 1985/1988), Lecture Notes in Math., vol. 1383, Springer, Berlin, 1989, pp. 89–110, DOI https://doi.org/10.1007/BFb0083572.
10.1007/BFb0083572
Google Scholar
28W. L. Hoyt, On twisted Legendre equations and Kummer surfaces, Theta functions—Bowdoin 1987, Part 1 (Brunswick, ME, 1987), Proc. Sympos. Pure Math. 49 (1989) 695–707.
10.1090/pspum/049.1/1013162
Google Scholar
29W. L. Hoyt and C. F. Schwartz, Yoshida surfaces with Picard number $\rho \ge 17$ , Proceedings on Moonshine and related topics (Montréal, QC, 1999), CRM Proc. Lecture Notes, vol. 30, pp. 71–78. Amer. Math. Soc., Providence, RI (2001).
10.1090/crmp/030/06
Google Scholar
30H. Inose, Defining equations of singular $K3$ surfaces and a notion of isogeny, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo, 1978, pp. 495–502. https://mathscinet.ams.org/mathscinet-getitem?mr=578868.
Google Scholar
31J. H. Keum, Automorphisms of Jacobian Kummer surfaces, Compositio Math. 107 (1997), no. 3, 269–288, DOI http://doi.org/10.1023/A:1000148907120.
10.1023/A:1000148907120
Web of Science® Google Scholar
32J. H. Keum, Automorphisms of a generic Jacobian Kummer surface, Geom. Dedicata 76 (1999), no. 2, 177–181, DOI http://doi.org/10.1023/A:1005019201955.
10.1023/A:1005019201955
Web of Science® Google Scholar
33Y. Kimura, Discrete gauge groups in certain F-theory models in six dimensions, J. High Energy Phys. 2019 (2019), no. 27. https://doi.org/10.1007/JHEP07(2019)027
10.1007/JHEP07(2019)027
Google Scholar
34Y. Kimura, Nongeometric heterotic strings and dual F-theory with enhanced gauge groups, J. High Energy Phys. 2019 (2019), no. 36. https://doi.org/10.1007/JHEP02(2019)036.
10.1007/JHEP02(2019)036
Google Scholar
35R. Kloosterman, Classification of all Jacobian elliptic fibrations on certain $K3$ surfaces, J. Math. Soc. Japan 58 (2006), no. 3, 665–680. https://mathscinet.ams.org/mathscinet-getitem?mr=2254405.
10.2969/jmsj/1156342032
Web of Science® Google Scholar
36K. Kodaira, On compact analytic surfaces. II, III, Ann. of Math. (2) 77 (1963), 563–626; ibid. 78 (1963), 1–40. https://mathscinet.ams.org/mathscinet-getitem?mr=0184257.
10.2307/1970131
Web of Science® Google Scholar
37S. Kondo, Algebraic $K3$ surfaces with finite automorphism groups, Nagoya Math. J. 116 (1989), 1–15. https://mathscinet.ams.org/mathscinet-getitem?mr=1029967.
10.1017/S0027763000001653
Web of Science® Google Scholar
38A. Kumar, $K3$ surfaces associated with curves of genus two, Int. Math. Res. Not. 2008 (2008), rnm165. http://doi.org/10.1093/imrn/rnm165.
10.1093/imrn/rnm165
Google Scholar
39A. Kumar, Elliptic fibrations on a generic Jacobian Kummer surface, J. Algebraic Geom. 23 (2014), no. 4, 599–667. https://mathscinet.ams.org/mathscinet-getitem?mr=3263663.
10.1090/S1056-3911-2014-00620-2
Web of Science® Google Scholar
40M. Kuwata and T. Shioda, Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface, K. Konno and V. Nguyen-Khac (Eds.), Algebraic geometry in East Asia—Hanoi 2005, Adv. Stud. Pure Math., vol. 50, Math. Soc. Japan, Tokyo, 2008, pp. 177–215, DOI https://doi.org/10.2969/aspm/05010177. https://mathscinet.ams.org/mathscinet-getitem?mr=2409557.
10.2969/aspm/05010177
Google Scholar
41G. Lombardo, C. Peters, and M. Schütt, Abelian fourfolds of Weil type and certain K3 double planes, Rend. Semin. Mat. Univ. Politec. Torino 71 (2013), no. 3-4, 339–383. https://mathscinet.ams.org/mathscinet-getitem?mr=3506391.
Google Scholar
42A. Malmendier, Kummer surfaces associated with Seiberg-Witten curves, J. Geom. Phys. 62 (2012), no. 1, 107–123, DOI http://doi.org/10.1016/j.geomphys.2011.09.010.
10.1016/j.geomphys.2011.09.010
Web of Science® Google Scholar
43A. Malmendier and D. R. Morrison, K3 surfaces, modular forms, and non-geometric heterotic compactifications, Lett. Math. Phys. 105 (2015), no. 8, 1085–1118, DOI http://doi.org/10.1007/s11005-015-0773-y.
10.1007/s11005-015-0773-y
Web of Science® Google Scholar
44A. Malmendier and T. Shaska, The Satake sextic in F-theory, J. Geom. Phys. 120 (2017), 290–305. https://mathscinet.ams.org/mathscinet-getitem?mr=3712162.
10.1016/j.geomphys.2017.06.010
Web of Science® Google Scholar
45A. Malmendier and Y. Sung, Counting rational points on Kummer surfaces, Res. Number Theory 5 (2019), no. 3, Paper No. 27, 23. https://mathscinet.ams.org/mathscinet-getitem?mr=3992148.
10.1007/s40993-019-0166-x
Web of Science® Google Scholar
46K. Matsumoto, Theta functions on the classical bounded symmetric domain of type ${\rm I}_{2,2}$ , Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), no. 1, 1–5. http://projecteuclid.org/euclid.pja/1195512258.
10.3792/pjaa.67.1
Google Scholar
47K. Matsumoto, Theta functions on the bounded symmetric domain of type $I_{2,2}$ and the period map of a 4-parameter family of $K3$ surfaces, Math. Ann. 295 (1993), no. 3, 383–409, DOI http://doi.org/10.1007/BF01444893.
10.1007/BF01444893
Google Scholar
48A. Mehran, Double covers of Kummer surfaces, Manuscripta Math. 123 (2007), no. 2, 205–235, DOI http://doi.org/10.1007/s00229-007-0092-4.
10.1007/s00229-007-0092-4
Web of Science® Google Scholar
49D. R. Morrison, On $K3$ surfaces with large Picard number, Invent. Math. 75 (1984), no. 1, 105–121, DOI http://doi.org/10.1007/BF01403093.
10.1007/BF01403093
Web of Science® Google Scholar
50A. Nagano and K. Ueda, The ring of modular forms of $\mathrm{O}(2,4;\mathbb {Z})$ with characters, Hokkaido Math. J. 51 (2022), no. 2, 275–286.
10.14492/hokmj/2020-355
Web of Science® Google Scholar
51V. V. Nikulin, An analogue of the Torelli theorem for Kummer surfaces of Jacobians, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 22–41. https://mathscinet.ams.org/mathscinet-getitem?mr=0357410.
Google Scholar
52V. V. Nikulin, Kummer surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 2, 278–293, 471. https://mathscinet.ams.org/mathscinet-getitem?mr=0429917.
Google Scholar
53V. V. Nikulin, Finite groups of automorphisms of Kählerian $K3$ surfaces, Trudy Moskov. Mat. Obshch. 38 (1979), 75–137. https://mathscinet.ams.org/mathscinet-getitem?mr=544937.
Google Scholar
54V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238. https://mathscinet.ams.org/mathscinet-getitem?mr=525944.
Google Scholar
55V. V. Nikulin, Quotient-groups of groups of automorphisms of hyperbolic forms of subgroups generated by 2-reflections, Dokl. Akad. Nauk SSSR 248 (1979), no. 6, 1307–1309. https://mathscinet.ams.org/mathscinet-getitem?mr=556762.
Google Scholar
56V. V. Nikulin, Quotient-groups of groups of automorphisms of hyperbolic forms by subgroups generated by 2-reflections, R. V. Gamkrelidze (Ed.), Algebro-geometric applications, Current problems in mathematics, Vol. 18, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981, pp. 3–114. https://mathscinet.ams.org/mathscinet-getitem?mr=633160.
Google Scholar
57V. V. Nikulin, Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. Algebrogeometric applications, J. Sov. Math. 22 (1983), no. 4, 1401–1475, https://doi.org/10.1007/BF01094757.
10.1007/BF01094757
Google Scholar
58V. V. Nikulin, $K3$ surfaces with a finite group of automorphisms and a Picard group of rank three, Algebraic geometry and its applications, Trudy Mat. Inst. Steklov. vol. 165, 1984, pp. 119–142. https://mathscinet.ams.org/mathscinet-getitem?mr=752938
Google Scholar
59V. V. Nikulin, Elliptic fibrations on $\rm K3$ surfaces, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 1, 253–267. https://mathscinet.ams.org/mathscinet-getitem?mr=3165023.
10.1017/S0013091513000953
Web of Science® Google Scholar
60K. Oguiso, On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curves, J. Math. Soc. Japan 41 (1989), no. 4, 651–680, DOI http://doi.org/10.2969/jmsj/04140651.
10.2969/jmsj/04140651
Web of Science® Google Scholar
61X. Roulleau, An atlas of K3 surfaces with finite automorphism group, Épijournal Géom. Algébrique 6 (2022), Art. 19, 95 p.
Google Scholar
62M. Schütt, Sandwich theorems for Shioda-Inose structures, Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), no. 1, 211–224. https://mathscinet.ams.org/mathscinet-getitem?mr=3087091.
Google Scholar
63I. Shimada, On elliptic $K3$ surfaces, Michigan Math. J. 47 (2000), no. 3, 423–446. https://mathscinet.ams.org/mathscinet-getitem?mr=1813537.
10.1307/mmj/1030132587
Web of Science® Google Scholar
64T. Shioda, Kummer sandwich theorem of certain elliptic $K3$ surfaces, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 8, 137–140. https://mathscinet.ams.org/mathscinet-getitem?mr=2279280.
Google Scholar
65H. Sterk, Finiteness results for algebraic $K3$ surfaces, Math. Z. 189 (1985), no. 4, 507–513. https://mathscinet.ams.org/mathscinet-getitem?mr=786280.
10.1007/BF01168156
Web of Science® Google Scholar
66B. van Geemen, An introduction to the Hodge conjecture for abelian varieties, A. Albano and F. Bardelli (Eds.), Algebraic cycles and Hodge theory (Torino, 1993), Lecture Notes in Math., vol. 1594, Springer, Berlin, 1994, pp. 233–252, DOI https://doi.org/10.1007/978-3-540-49046-3_5. https://mathscinet.ams.org/mathscinet-getitem?mr=1335243.
10.1007/978-3-540-49046-3_5
Google Scholar
67B. van Geemen and A. Sarti, Nikulin involutions on $K3$ surfaces, Math. Z. 255 (2007), no. 4, 731–753. https://mathscinet.ams.org/mathscinet-getitem?mr=2274533.
10.1007/s00209-006-0047-6
Web of Science® Google Scholar
68E. B. Vinberg, On automorphic forms on symmetric domains of type IV, Uspekhi Mat. Nauk 65 (2010), no. 3(393), 193–194. https://mathscinet.ams.org/mathscinet-getitem?mr=2682724.
Google Scholar
69E. B. Vinberg, Some free algebras of automorphic forms on symmetric domains of type IV, Transform. Groups 15 (2010), no. 3, 701–741. https://mathscinet.ams.org/mathscinet-getitem?mr=2718942.
10.1007/s00031-010-9107-4
Web of Science® Google Scholar
70E. B. Vinberg, On the algebra of Siegel modular forms of genus 2, Trans. Moscow Math. Soc. (2013), 1–13. https://doi.org/10.1090/S0077-1554-2014-00217-X.
10.1090/S0077-1554-2014-00217-X
Google Scholar

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