The Hodge standard conjecture for self-products of K3 surfaces
Authors:
Kazuhiro Ito, Tetsushi Ito and Teruhisa Koshikawa
Journal:
J. Algebraic Geom.
DOI:
https://doi.org/10.1090/jag/840
Published electronically:
October 3, 2024
Full-text PDF
Abstract |
References |
Additional Information
Abstract: As an application of our previous work on CM liftings of K3 surfaces and the Tate conjecture, we prove the Hodge standard conjecture for squares of K3 surfaces. We also deduce the Hodge standard conjecture for all the powers of certain K3 surfaces.
References
- Giuseppe Ancona, Standard conjectures for abelian fourfolds, Invent. Math. 223 (2021), no. 1, 149–212. MR 4199442, DOI 10.1007/s00222-020-00990-7
- Yves André, Pour une théorie inconditionnelle des motifs, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 5–49 (French). MR 1423019
- Yves André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses [Panoramas and Syntheses], vol. 17, Société Mathématique de France, Paris, 2004 (French, with English and French summaries). MR 2115000
- M. Artin, Supersingular $K3$ surfaces, Ann. Sci. École Norm. Sup. (4) 7 (1974), 543–567 (1975). MR 371899
- M. Artin and B. Mazur, Formal groups arising from algebraic varieties, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 1, 87–131. MR 457458
- Nikolay Buskin, Every rational Hodge isometry between two $K3$ surfaces is algebraic, J. Reine Angew. Math. 755 (2019), 127–150. MR 4015230, DOI 10.1515/crelle-2017-0027
- L. Clozel, Equivalence numérique et équivalence cohomologique pour les variétés abéliennes sur les corps finis, Ann. of Math. (2) 150 (1999), no. 1, 151–163 (French). MR 1715322, DOI 10.2307/121099
- A. Grothendieck, Standard conjectures on algebraic cycles, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) Tata Inst. Fundam. Res. Stud. Math., vol. 4, Tata Inst. Fund. Res., Bombay, 1969, pp. 193–199. MR 268189
- Daniel Huybrechts, Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, vol. 158, Cambridge University Press, Cambridge, 2016. MR 3586372, DOI 10.1017/CBO9781316594193
- Daniel Huybrechts, Motives of isogenous K3 surfaces, Comment. Math. Helv. 94 (2019), no. 3, 445–458. MR 4014777, DOI 10.4171/CMH/465
- Kazuhiro Ito, Unconditional construction of $K3$ surfaces over finite fields with given $L$-function in large characteristic, Manuscripta Math. 159 (2019), no. 3-4, 281–300. MR 3959263, DOI 10.1007/s00229-018-1066-4
- Kazuhiro Ito, On the supersingular reduction of $K3$ surfaces with complex multiplication, Int. Math. Res. Not. IMRN 20 (2020), 7306–7346. MR 4172684, DOI 10.1093/imrn/rny210
- Kazuhiro Ito, Tetsushi Ito, and Teruhisa Koshikawa, CM liftings of $K3$ surfaces over finite fields and their applications to the Tate conjecture, Forum Math. Sigma 9 (2021), Paper No. e29, 70. MR 4241794, DOI 10.1017/fms.2021.24
- Tetsushi Ito, Weight-monodromy conjecture for $p$-adically uniformized varieties, Invent. Math. 159 (2005), no. 3, 607–656. MR 2125735, DOI 10.1007/s00222-004-0395-y
- Uwe Jannsen, Motives, numerical equivalence, and semi-simplicity, Invent. Math. 107 (1992), no. 3, 447–452. MR 1150598, DOI 10.1007/BF01231898
- Bruno Kahn, Jacob P. Murre, and Claudio Pedrini, On the transcendental part of the motive of a surface, Algebraic cycles and motives. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 344, Cambridge Univ. Press, Cambridge, 2007, pp. 143–202. MR 2187153
- Nicholas M. Katz and William Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77. MR 332791, DOI 10.1007/BF01405203
- Wansu Kim and Keerthi Madapusi Pera, 2-adic integral canonical models, Forum Math. Sigma 4 (2016), Paper No. e28, 34. MR 3569319, DOI 10.1017/fms.2016.23
- Mark Kisin, $\textrm {mod}\,p$ points on Shimura varieties of abelian type, J. Amer. Math. Soc. 30 (2017), no. 3, 819–914. MR 3630089, DOI 10.1090/jams/867
- S. L. Kleiman, Algebraic cycles and the Weil conjectures, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 359–386. MR 292838
- Steven L. Kleiman, The standard conjectures, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 3–20. MR 1265519, DOI 10.1090/pspum/055.1/1265519
- Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779, DOI 10.1090/coll/044
- Teruhisa Koshikawa, The numerical Hodge standard conjecture for the square of a simple abelian variety of prime dimension, Manuscripta Math. 173 (2024), no. 3-4, 1161–1169. MR 4704771, DOI 10.1007/s00229-023-01482-7
- Christian Liedtke, Lectures on supersingular K3 surfaces and the crystalline Torelli theorem, K3 surfaces and their moduli, Progr. Math., vol. 315, Birkhäuser/Springer, [Cham], 2016, pp. 171–235. MR 3524169, DOI 10.1007/978-3-319-29959-4_{8}
- Keerthi Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic, Invent. Math. 201 (2015), no. 2, 625–668. MR 3370622, DOI 10.1007/s00222-014-0557-5
- Keerthi Madapusi Pera, Erratum to appendix to ‘2-adic integral canonical models’, Forum Math. Sigma 8 (2020), Paper No. e14, 11. MR 4076653, DOI 10.1017/fms.2020.2
- Eyal Markman, The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians, J. Eur. Math. Soc. (JEMS) 25 (2023), no. 1, 231–321. MR 4556784, DOI 10.4171/jems/1199
- J. S. Milne, Lefschetz motives and the Tate conjecture, Compositio Math. 117 (1999), no. 1, 45–76. MR 1692999, DOI 10.1023/A:1000776613765
- J. S. Milne, Polarizations and Grothendieck’s standard conjectures, Ann. of Math. (2) 155 (2002), no. 2, 599–610. MR 1906596, DOI 10.2307/3062126
- J. S. Milne, On the Tate and standard conjectures over finite fields, arXiv:1907.04143 (2019).
- J. S. Milne, The Tate and standard conjectures for certain abelian varieties, arXiv:2112.12815 (2022).
- I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, The arithmetic of surfaces of type $\textrm {K}3$, Trudy Mat. Inst. Steklov. 132 (1973), 44–54, 264 (Russian). MR 335521
- Lenny Taelman, K3 surfaces over finite fields with given $L$-function, Algebra Number Theory 10 (2016), no. 5, 1133–1146. MR 3531364, DOI 10.2140/ant.2016.10.1133
- John Tate, Conjectures on algebraic cycles in $l$-adic cohomology, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 71–83. MR 1265523, DOI 10.1090/pspum/055.1/1265523
- Z. Yang, The Tate conjecture for motivic endomorphisms of K3 surfaces over finite fields, arXiv:2110.01350 (2021).
- Jeng-Daw Yu and Noriko Yui, $K3$ surfaces of finite height over finite fields, J. Math. Kyoto Univ. 48 (2008), no. 3, 499–519. MR 2511048, DOI 10.1215/kjm/1250271381
- Zhiwei Yun and Wei Zhang, Shtukas and the Taylor expansion of $L$-functions, Ann. of Math. (2) 186 (2017), no. 3, 767–911. MR 3702678, DOI 10.4007/annals.2017.186.3.2
- Yu. G. Zarhin, Hodge groups of $K3$ surfaces, J. Reine Angew. Math. 341 (1983), 193–220. MR 697317, DOI 10.1515/crll.1983.341.193
- Yuri G. Zarhin, Transcendental cycles on ordinary $K3$ surfaces over finite fields, Duke Math. J. 72 (1993), no. 1, 65–83. MR 1242879, DOI 10.1215/S0012-7094-93-07203-1
- Yu. G. Zarhin, The Tate conjecture for nonsimple abelian varieties over finite fields, Algebra and number theory (Essen, 1992) de Gruyter, Berlin, 1994, pp. 267–296. MR 1285371
- Yuri G. Zarhin, The Tate conjecture for powers of ordinary $K3$ surfaces over finite fields, J. Algebraic Geom. 5 (1996), no. 1, 151–172. MR 1358039
- Yuri G. Zarhin, Eigenvalues of Frobenius endomorphisms of abelian varieties of low dimension, J. Pure Appl. Algebra 219 (2015), no. 6, 2076–2098. MR 3299720, DOI 10.1016/j.jpaa.2014.07.024
References
- Giuseppe Ancona, Standard conjectures for abelian fourfolds, Invent. Math. 223 (2021), no. 1, 149–212. MR 4199442, DOI 10.1007/s00222-020-00990-7
- Yves André, Pour une théorie inconditionnelle des motifs, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 5–49 (French). MR 1423019
- Yves André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses [Panoramas and Syntheses], vol. 17, Société Mathématique de France, Paris, 2004 (French, with English and French summaries). MR 2115000
- M. Artin, Supersingular $K3$ surfaces, Ann. Sci. École Norm. Sup. (4) 7 (1974), 543–567 (1975). MR 371899
- M. Artin and B. Mazur, Formal groups arising from algebraic varieties, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 1, 87–131. MR 457458
- Nikolay Buskin, Every rational Hodge isometry between two $K3$ surfaces is algebraic, J. Reine Angew. Math. 755 (2019), 127–150. MR 4015230, DOI 10.1515/crelle-2017-0027
- L. Clozel, Equivalence numérique et équivalence cohomologique pour les variétés abéliennes sur les corps finis, Ann. of Math. (2) 150 (1999), no. 1, 151–163 (French). MR 1715322, DOI 10.2307/121099
- A. Grothendieck, Standard conjectures on algebraic cycles, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) Tata Inst. Fundam. Res. Stud. Math., vol. 4, Tata Inst. Fund. Res., Bombay, 1969, pp. 193–199. MR 268189
- Daniel Huybrechts, Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, vol. 158, Cambridge University Press, Cambridge, 2016. MR 3586372, DOI 10.1017/CBO9781316594193
- Daniel Huybrechts, Motives of isogenous K3 surfaces, Comment. Math. Helv. 94 (2019), no. 3, 445–458. MR 4014777, DOI 10.4171/CMH/465
- Kazuhiro Ito, Unconditional construction of $K3$ surfaces over finite fields with given $L$-function in large characteristic, Manuscripta Math. 159 (2019), no. 3-4, 281–300. MR 3959263, DOI 10.1007/s00229-018-1066-4
- Kazuhiro Ito, On the supersingular reduction of $K3$ surfaces with complex multiplication, Int. Math. Res. Not. IMRN 20 (2020), 7306–7346. MR 4172684, DOI 10.1093/imrn/rny210
- Kazuhiro Ito, Tetsushi Ito, and Teruhisa Koshikawa, CM liftings of $K3$ surfaces over finite fields and their applications to the Tate conjecture, Forum Math. Sigma 9 (2021), Paper No. e29, 70. MR 4241794, DOI 10.1017/fms.2021.24
- Tetsushi Ito, Weight-monodromy conjecture for $p$-adically uniformized varieties, Invent. Math. 159 (2005), no. 3, 607–656. MR 2125735, DOI 10.1007/s00222-004-0395-y
- Uwe Jannsen, Motives, numerical equivalence, and semi-simplicity, Invent. Math. 107 (1992), no. 3, 447–452. MR 1150598, DOI 10.1007/BF01231898
- Bruno Kahn, Jacob P. Murre, and Claudio Pedrini, On the transcendental part of the motive of a surface, Algebraic cycles and motives. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 344, Cambridge Univ. Press, Cambridge, 2007, pp. 143–202. MR 2187153
- Nicholas M. Katz and William Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77. MR 332791, DOI 10.1007/BF01405203
- Wansu Kim and Keerthi Madapusi Pera, 2-adic integral canonical models, Forum Math. Sigma 4 (2016), Paper No. e28, 34. MR 3569319, DOI 10.1017/fms.2016.23
- Mark Kisin, $\operatorname {mod}p$ points on Shimura varieties of abelian type, J. Amer. Math. Soc. 30 (2017), no. 3, 819–914. MR 3630089, DOI 10.1090/jams/867
- S. L. Kleiman, Algebraic cycles and the Weil conjectures, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 359–386. MR 292838
- Steven L. Kleiman, The standard conjectures, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 3–20. MR 1265519, DOI 10.1090/pspum/055.1/1265519
- Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779, DOI 10.1090/coll/044
- Teruhisa Koshikawa, The numerical Hodge standard conjecture for the square of a simple abelian variety of prime dimension, Manuscripta Math. 173 (2024), no. 3-4, 1161–1169. MR 4704771, DOI 10.1007/s00229-023-01482-7
- Christian Liedtke, Lectures on supersingular K3 surfaces and the crystalline Torelli theorem, K3 surfaces and their moduli, Progr. Math., vol. 315, Birkhäuser/Springer, [Cham], 2016, pp. 171–235. MR 3524169, DOI 10.1007/978-3-319-29959-4_8
- Keerthi Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic, Invent. Math. 201 (2015), no. 2, 625–668. MR 3370622, DOI 10.1007/s00222-014-0557-5
- Keerthi Madapusi Pera, Erratum to appendix to ‘2-adic integral canonical models’, Forum Math. Sigma 8 (2020), Paper No. e14, 11. MR 4076653, DOI 10.1017/fms.2020.2
- Eyal Markman, The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians, J. Eur. Math. Soc. (JEMS) 25 (2023), no. 1, 231–321. MR 4556784, DOI 10.4171/jems/1199
- J. S. Milne, Lefschetz motives and the Tate conjecture, Compositio Math. 117 (1999), no. 1, 45–76. MR 1692999, DOI 10.1023/A:1000776613765
- J. S. Milne, Polarizations and Grothendieck’s standard conjectures, Ann. of Math. (2) 155 (2002), no. 2, 599–610. MR 1906596, DOI 10.2307/3062126
- J. S. Milne, On the Tate and standard conjectures over finite fields, arXiv:1907.04143 (2019).
- J. S. Milne, The Tate and standard conjectures for certain abelian varieties, arXiv:2112.12815 (2022).
- I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, The arithmetic of surfaces of type $\mathrm {K}3$, Trudy Mat. Inst. Steklov. 132 (1973), 44–54, 264 (Russian). MR 335521
- Lenny Taelman, K3 surfaces over finite fields with given $L$-function, Algebra Number Theory 10 (2016), no. 5, 1133–1146. MR 3531364, DOI 10.2140/ant.2016.10.1133
- John Tate, Conjectures on algebraic cycles in $l$-adic cohomology, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 71–83. MR 1265523, DOI 10.1090/pspum/055.1/1265523
- Z. Yang, The Tate conjecture for motivic endomorphisms of K3 surfaces over finite fields, arXiv:2110.01350 (2021).
- Jeng-Daw Yu and Noriko Yui, $K3$ surfaces of finite height over finite fields, J. Math. Kyoto Univ. 48 (2008), no. 3, 499–519. MR 2511048, DOI 10.1215/kjm/1250271381
- Zhiwei Yun and Wei Zhang, Shtukas and the Taylor expansion of $L$-functions, Ann. of Math. (2) 186 (2017), no. 3, 767–911. MR 3702678, DOI 10.4007/annals.2017.186.3.2
- Yu. G. Zarhin, Hodge groups of $K3$ surfaces, J. Reine Angew. Math. 341 (1983), 193–220. MR 697317, DOI 10.1515/crll.1983.341.193
- Yuri G. Zarhin, Transcendental cycles on ordinary $K3$ surfaces over finite fields, Duke Math. J. 72 (1993), no. 1, 65–83. MR 1242879, DOI 10.1215/S0012-7094-93-07203-1
- Yu. G. Zarhin, The Tate conjecture for nonsimple abelian varieties over finite fields, Algebra and number theory (Essen, 1992) de Gruyter, Berlin, 1994, pp. 267–296. MR 1285371
- Yuri G. Zarhin, The Tate conjecture for powers of ordinary $K3$ surfaces over finite fields, J. Algebraic Geom. 5 (1996), no. 1, 151–172. MR 1358039
- Yuri G. Zarhin, Eigenvalues of Frobenius endomorphisms of abelian varieties of low dimension, J. Pure Appl. Algebra 219 (2015), no. 6, 2076–2098. MR 3299720, DOI 10.1016/j.jpaa.2014.07.024
Additional Information
Kazuhiro Ito
Affiliation:
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
MR Author ID:
1260950
Email:
kazuhiro.ito.c3@tohoku.ac.jp
Tetsushi Ito
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
MR Author ID:
684347
ORCID:
0000-0002-6199-0218
Email:
tetsushi@math.kyoto-u.ac.jp
Teruhisa Koshikawa
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
MR Author ID:
1253617
Email:
teruhisa@kurims.kyoto-u.ac.jp
Received by editor(s):
December 2, 2022
Received by editor(s) in revised form:
May 2, 2024
Published electronically:
October 3, 2024
Additional Notes:
The work of the first author was supported by JSPS KAKENHI Grant Number 22K20332. The work of the third author was supported by JSPS KAKENHI Grant Number 20K14284. The work of the second and the third author was supported by JSPS KAKENHI Grant Number 21H00973
Article copyright:
© Copyright 2024
University Press, Inc.
