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Standard conjectures for abelian fourfolds. (English) Zbl 1467.14023

Summary: Let \(A\) be an abelian fourfold in characteristic \(p\). We prove the standard conjecture of Hodge type for \(A\), namely that the intersection product \[ \mathcal{Z}^2_\text{num} (A)_\mathbb{Q} \times \mathcal{Z}_\text{num}^2 (A)_\mathbb{Q} \longrightarrow \mathbb{Q} \] is of signature \((\rho_2 - \rho_1 +1; \rho_1 - 1)\), with \(\rho_n = \dim \mathcal{Z}_\text{num}^n (A)_\mathbb{Q}.\) (Equivalently, it is positive definite when restricted to primitive classes for any choice of the polarization.) The approach consists in reformulating this question into a \(p\)-adic problem and then using \(p\)-adic Hodge theory to solve it. By combining this result with a theorem of Clozel we deduce that numerical equivalence on \(A\) coincides with \(\ell\)-adic homological equivalence on \(A\) for infinitely many prime numbers \(\ell\). Hence, what is missing among the standard conjectures for abelian fourfolds is \(\ell\)-independency of \(\ell\)-adic homological equivalence.

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
11E12 Quadratic forms over global rings and fields
11S37 Langlands-Weil conjectures, nonabelian class field theory
14C15 (Equivariant) Chow groups and rings; motives
14L15 Group schemes
14K99 Abelian varieties and schemes
14J35 \(4\)-folds
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[1] Ancona, G.; Huber, A.; Lehalleur, SP, On the relative motive of a commutative group scheme, Algebra Geom., 3, 2, 150-178 (2016) · Zbl 1354.14011
[2] André, Y.; Kahn, B., Nilpotence, radicaux et structures monoïdales, Rend. Sem. Mat. Univ. Padova, 108, 107-291 (2002) · Zbl 1165.18300
[3] Ancona, G., On the standard conjecture of Hodge type for abelian varieties, Oberwolfach Rep., 31, 2016, 1771-1773 (2016)
[4] Ancona, G.: Conservativity of realization functors on motives of abelian type over finite fields. In: Fresán, J., Jossen, Pink (eds.) Proceedings of “Motives and Complex Multiplication” to appear
[5] André, Y.: Une introduction aux motifs (motifs purs, motifs mixtes, périodes). In: Panoramas et Synthèses [Panoramas and Syntheses], vol. 17. Société Mathématique de France, Paris (2004) · Zbl 1060.14001
[6] Beauville, A., Some surfaces with maximal Picard number, J. École Polytech. Math., 1, 101-116 (2014) · Zbl 1326.14080
[7] Bini, G., Laterveer, R., Pacienza, G.: Voisin’s conjecture for zero-cycles on Calabi-Yau varieties and their mirrors. Adv. Geom. 20(1), 91-108 (2020) · Zbl 1437.14015
[8] Berthelot, P.; Ogus, A., \(F\)-isocrystals and de Rham cohomology, Invent. Math., 72, 2, 159-199 (1983) · Zbl 0516.14017
[9] Bronowski, J., Curves whose grade is not positive in the theory of the base, J. Lond. Math. Soc., 13, 2, 86-90 (1938) · JFM 64.0687.03
[10] Colmez, P.; Fontaine, J-M, Construction des représentations \(p\)-adiques semi-stables, Invent. Math., 140, 1, 1-43 (2000) · Zbl 1010.14004
[11] Clozel, L., Equivalence numérique et équivalence cohomologique pour les variétés abéliennes sur les corps finis, Ann. Math. (2), 150, 1, 151-163 (1999) · Zbl 0995.14018
[12] Colmez, P., Espaces de Banach de dimension finie, J. Inst. Math. Jussieu, 1, 3, 331-439 (2002) · Zbl 1044.11102
[13] Deninger, C.; Murre, J., Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math., 422, 201-219 (1991) · Zbl 0745.14003
[14] Déglise, F.; Nizioł, W., On \(p\)-adic absolute Hodge cohomology and syntomic coefficients. I, Comment. Math. Helv., 93, 1, 71-131 (2018) · Zbl 1423.14144
[15] Faltings, G.; Kashiwara, M., Crystalline cohomology and \(p\)-adic Galois-representations, Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988), 25-80 (1989), Baltimore: Johns Hopkins University Press, Baltimore · Zbl 0805.14008
[16] Fontaine, J.-M., Messing, W.: \(p\)-adic periods and \(p\)-adic étale cohomology. In: Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985), Volume 67 of Contemporary Mathematics, pp. 179-207. American Mathematical Society, Providence, RI (1987) · Zbl 0632.14016
[17] Fontaine, J-M, Sur certains types de représentations \(p\)-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. Math. (2), 115, 3, 529-577 (1982) · Zbl 0544.14016
[18] Fontaine, J-M, Le corps des périodes \(p\)-adiques, Astérisque, 223, 59-111 (1994) · Zbl 0940.14012
[19] Grothendieck, A., Sur une note de Mattuck-Tate, J. Reine Angew. Math., 200, 208-215 (1958) · Zbl 0084.16904
[20] Grothendieck, A.: Standard conjectures on algebraic cycles. In: Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), pp. 193-199. Oxford University Press, London (1969) · Zbl 0201.23301
[21] Gillet, H., Soulé, C.: Arithmetic analogs of the standard conjectures. In: Motives (Seattle, WA, 1991), Volume 55 of Proceedings of Symposium on Pure Mathematics, pp. 129-140. American Mathematical Society, Providence, RI (1994) · Zbl 0820.14007
[22] Hyodo, O.; Kato, K., Semi-stable reduction and crystalline cohomology with logarithmic poles, Astérisque, 223, 221-268 (1994) · Zbl 0852.14004
[23] Hodge, WVD, The geometric genus of a surface as a topological invariant, J. Lond. Math. Soc., 8, 4, 312-319 (1933) · JFM 59.1313.03
[24] Honda, T., Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Jpn., 20, 83-95 (1968) · Zbl 0203.53302
[25] Ireland, K.F., Rosen, M.I.: A classical introduction to modern number theory. In: Graduate Texts in Mathematics, vol. 84. Springer, New York, Berlin (1982). (Revised edition of ıt Elements of Number Theory) · Zbl 0482.10001
[26] Jannsen, U., Motives, numerical equivalence, and semi-simplicity, Invent. Math., 107, 3, 447-452 (1992) · Zbl 0762.14003
[27] Jannsen, U.: On finite-dimensional motives and Murre’s conjecture. In: Algebraic Cycles and Motives, Volume 344 of London Mathematical Society, Lecture Note Series, vol. 2, pp. 112-142. Cambridge University Press, Cambridge (2007) · Zbl 1127.14007
[28] Kings, G., Higher regulators, Hilbert modular surfaces, and special values of \(L\)-functions, Duke Math. J., 92, 1, 61-127 (1998) · Zbl 0962.11024
[29] Kleiman, S.: Algebraic cycles and the Weil conjectures. In: Dix esposés sur la cohomologie des schémas, pp. 359-386. North-Holland, Amsterdam (1968) · Zbl 0198.25902
[30] Künnemann, K., Maillot, V.: Théorèmes de Lefschetz et de Hodge arithmétiques pour les variétés admettant une décomposition cellulaire. In: Regulators in Analysis, Geometry and Number Theory, Volume 171 of Progress in Mathematics, pp. 197-205. Birkhäuser Boston, Boston, MA (2000) · Zbl 1083.14025
[31] Kresch, A.; Tamvakis, H., Standard conjectures for the arithmetic Grassmannian \(G(2, N)\) and Racah polynomials, Duke Math. J., 110, 2, 359-376 (2001) · Zbl 1072.14514
[32] Künnemann, K., A Lefschetz decomposition for chow motives of abelian schemes, Invent. Math., 113, 4, 85-102 (1993) · Zbl 0806.14001
[33] Künnemann, K.: On the Chow motive of an abelian scheme. In: Motives, Volume 55.1 of Proceedings of Symposia in Pure Mathematics, pp. 189-205. American Mathematical Society (1994) · Zbl 0823.14032
[34] Künnemann, K., Some remarks on the arithmetic Hodge index conjecture, Compos. Math., 99, 2, 109-128 (1995) · Zbl 0845.14006
[35] Künnemann, K., The Kähler identity for bigraded Hodge-Lefschetz modules and its application in non-Archimedean Arakelov geometry, J. Algebr. Geom., 7, 4, 651-672 (1998) · Zbl 0954.14017
[36] Lieberman, DI, Numerical and homological equivalence of algebraic cycles on Hodge manifolds, Am. J. Math., 90, 366-374 (1968) · Zbl 0159.50501
[37] Lenstra, HW Jr; Oort, F., Simple abelian varieties having a prescribed formal isogeny type, J. Pure Appl. Algebra, 4, 47-53 (1974) · Zbl 0279.14009
[38] Langlands, R.; Rapoport, M., Shimuravarietäten und Gerben, J. Reine Angew. Math., 378, 113-220 (1987) · Zbl 0615.14014
[39] Matsusaka, T., The criteria for algebraic equivalence and the torsion group, Am. J. Math., 79, 53-66 (1957) · Zbl 0077.34303
[40] Milne, JS, Lefschetz classes on abelian varieties, Duke Math. J., 96, 3, 639-675 (1999) · Zbl 0976.14009
[41] Milne, JS, Polarizations and Grothendieck’s standard conjectures, Ann. Math. (2), 155, 2, 599-610 (2002) · Zbl 1055.14008
[42] Mattuck, A.; Tate, J., On the inequality of Castelnuovo-Severi, Abh. Math. Sem. Univ. Hambg., 22, 295-299 (1958) · Zbl 0081.37604
[43] Mumford, D.: Abelian varieties. In: Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Published for the Tata Institute of Fundamental Research, Bombay (2008). (With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition)
[44] Moonen, BJJ; Zarhin, Y., Hodge classes and Tate classes on simple abelian fourfolds, Duke Math. J., 77, 3, 553-581 (1995) · Zbl 0874.14034
[45] Nizioł, W., On uniqueness of \(p\)-adic period morphisms, Pure Appl. Math. Q., 5, 1, 163-212 (2009) · Zbl 1200.14037
[46] O’Sullivan, P., Algebraic cycles on an abelian variety, J. Reine Angew. Math., 654, 1-81 (2011) · Zbl 1258.14006
[47] Rivano, NS, Catégories tannakiennes, Bull. Soc. Math. France, 100, 417-430 (1972) · Zbl 0246.14003
[48] Saito, M.: Monodromy filtration and positivity, pp. 1-19. preprint available at https://arxiv.org/pdf/math/0006162.pdf
[49] Segre, B., Intorno ad un teorema di Hodge sulla teoria della base per le curve di una superficie algebrica, Ann. Mat. Pura Appl., 16, 1, 157-163 (1937) · JFM 63.0622.01
[50] Serre, J.-P.: Cours d’arithmétique. Presses Universitaires de France, Paris, Deuxième édition revue et corrigée (1977), Le Mathématicien, No 2 · Zbl 0376.12001
[51] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. In: Kanô Memorial Lectures, No. 1. Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, Princeton University Press, Princeton, NJ (1971) · Zbl 0221.10029
[52] Shioda, T., The Hodge conjecture for Fermat varieties, Math. Ann., 245, 2, 175-184 (1979) · Zbl 0403.14007
[53] Shioda, T.; Katsura, T., On Fermat varieties, Tôhoku Math. J. (2), 31, 1, 97-115 (1979) · Zbl 0415.14022
[54] Tate, J., Endomorphisms of abelian varieties over finite fields, Invent. Math., 2, 134-144 (1966) · Zbl 0147.20303
[55] Tate, J.: Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda). In: Séminaire Bourbaki, vol. 1968/69: Exposés 347-363, Volume 175 of Lecture Notes in Mathematics, Exp. No. 352, pp. 95-110. Springer, Berlin (1971) · Zbl 0212.25702
[56] Zarhin, Y., Eigenvalues of Frobenius endomorphisms of abelian varieties of low dimension, J. Pure Appl. Algebra, 219, 6, 2076-2098 (2015) · Zbl 1397.11115
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