Gregory, Oliver; Liedtke, Christian

\(p\)-Adic Tate Conjectures and Abeloid Varieties

Doc. Math. 24, 1879-1934 (2019)
DOI: 10.25537/dm.2019v24.1879-1934
Communicated by Thomas Geisser

Summary

We explore Tate-type conjectures over \(p\)-adic fields, especially a conjecture of \textit{W. Raskind} [in: Algebra and number theory. Proceedings of the silver jubilee conference, Hyderabad, India, December 11--16, 2003. New Delhi: Hindustan Book Agency. 99--115 (2005; Zbl 1085.14009)] that predicts the surjectivity of \[(\text{NS}(X_{\overline{K}}) \otimes_{\mathbb{Z}}\mathbb{Q}_p)^{G_{K}}\longrightarrow H_{\text{ét}}^2(X_{\overline{K}},\mathbb{Q}_p(1))^{G_{K}}\] if \(X\) is smooth and projective over a \(p\)-adic field \(K\) and has totally degenerate reduction. Sometimes, this is related to \(p\)-adic uniformisation. For abelian varieties, Raskind's conjecture is equivalent to the question whether \[\text{Hom}(A,B)\otimes\mathbb{Q}_p \,\to\, \text{Hom}_{G_{K}}(V_p(A),V_p(B))\] is surjective if \(A\) and \(B\) are abeloid varieties over a \(p\)-adic field. \par Using \(p\)-adic Hodge theory and Fontaine's functors, we reformulate both problems into questions about the interplay of \(\mathbb{Q}\)-versus \(\mathbb{Q}_p\)-structures inside filtered \((\varphi,N)\)-modules. Finally, we disprove all of these conjectures and questions by showing that they can fail for algebraisable abeloid surfaces, that is, for abelian surfaces with totally degenerate reduction.

Mathematics Subject Classification

14F30, 11F80, 14C22, 14K02

Keywords/Phrases

Tate conjecture, abelian and abeloid varieties, \(p\)-adic fields and \(p\)-adic uniformisation, \(p\)-adic Hodge theory, filtered \((\varphi, N)\)-module, totally degenerate reduction

References

  • 1. Y. André, On the Shafarevich and Tate conjectures for hyper-Kähler varieties, Math. Ann. 305 (1996), no. 2, 205-248. DOI 10.1007/BF01444219; zbl 0942.14018; MR1391213.
  • 2. Y. André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synth\``eses 17, Soci\''eté Mathématique de France, 2004. zbl 1060.14001; MR2115000.
  • 3. L. Berger, An introduction to the theory of \(p\)-adic representations, Geometric aspects of Dwork theory, Vol. I, II, 255-292, Walter de Gruyter (2004). zbl 1118.11028; MR2023292; arxiv math/0210184.
  • 4. P. Berthelot, A. Ogus, \(F\)-isocrystals and de Rham cohomology. I, Invent. Math. 72 (1983), no. 2, 159-199. DOI 10.1007/BF01389319; zbl 0516.14017; MR0700767.
  • 5. P. Berthelot, Finitude et pureté cohomologique en cohomologie rigide, Invent. Math. 128 (1997), no. 2, 329-377. DOI 10.1007/s002220050143; zbl 0908.14005; MR1440308.
  • 6. A. Besser, E. de Shalit, \( \mathcal L\)-invariants of \(p\)-adically uniformized varieties, Ann. Math. Qué. 40 (2016), no. 1, 29-54. DOI 10.1007/s40316-015-0047-1; zbl 1375.14074; MR3512522.
  • 7. S. Bloch, H. Esnault, M. Kerz, \(p\)-adic deformation of algebraic cycle classes, Invent. Math. 195 (2014), no. 3, 673-722. DOI 10.1007/s00222-013-0461-4; zbl 1301.19005; MR3166216; arxiv 1203.2776.
  • 8. S. Bloch, H. Gillet, C. Soulé, Algebraic cycles on degenerate fibers, Arithmetic geometry (Cortona, 1994), 45-69, Sympos. Math., XXXVII, Cambridge University Press (1997). zbl 0955.14007; MR1472491.
  • 9. S. Bosch, W. L\``utkebohmert, M. Raynaud, N\''eron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21. Springer (1990). zbl 0705.14001; MR1045822.
  • 10. F. Charles, The Tate conjecture for K3 surfaces over finite fields, Invent. Math. 194 (2013), no. 1, 119-145. DOI 10.1007/s00222-012-0443-y; zbl 1282.14014; MR3103257; arxiv 1206.4002.
  • 11. B. Chiarellotto, C. Lazda, Combinatorial degenerations of surfaces and Calabi-Yau threefolds, Algebra Number Theory 10 (2016), no. 10, 2235-2266. DOI 10.2140/ant.2016.10.2235; zbl 1354.14057; MR3582018; arxiv 1602.04063.
  • 12. B. Chiarellotto, N. Tsuzuki, Clemens-Schmid exact sequence in characteristic \(p\), Math. Ann. 358 (2014), no. 3-4, 971-1004. DOI 10.1007/s00208-013-0980-8; zbl 1330.14027; MR3175147; arxiv 1111.0779.
  • 13. R. Coleman, The monodromy pairing, Asian J. Math. 4 (2000), no. 2, 315-330. DOI 10.4310/AJM.2000.v4.n2.a2; zbl 1092.14511; MR1797583.
  • 14. R. Coleman, A. Iovita, The Frobenius and monodromy operators for curves and abelian varieties, Duke Math. J. 97 (1999), no. 1, 171-215. DOI 10.1215/S0012-7094-99-09708-9; zbl 0962.14030; MR1682268; arxiv math/9701229.
  • 15. P. Colmez, J.-M. Fontaine, Construction des represéntations \(p\)-adiques semi-stables, Invent. Math. 140 (2000), no. 1, 1-43. DOI 10.1007/s002220000042; zbl 1010.14004; MR1779803.
  • 16. P. Colmez, W. Nizioł, Syntomic complexes and \(p\)-adic nearby cycles, Invent. Math. 208 (2017), no. 1, 1-108. DOI 10.1007/s00222-016-0683-3; zbl 1395.14013; MR3621832; arxiv 1505.06471.
  • 17. C. Consani, Double complexes and Euler \(L\)-factors, Compositio Math. 111 (1998), no. 3, 323-358. DOI 10.1023/A:1000362027455; zbl 0932.14011; MR1617133.
  • 18. S. Dasgupta, J. Teitelbaum, The \(p\)-adic upper half plane, p-adic geometry, 65-121, Univ. Lecture Series 45, Amer. Math. Soc. 2008. zbl 1153.14021; MR2482346.
  • 19. P. Deligne, Cristaux ordinaires et coordonnées canoniques, With the collaboration of L. Illusie. With an appendix by Nicholas M. Katz. Lecture Notes in Math., 868, Algebraic surfaces (Orsay, 1976-78), 80-137, Springer, Berlin-New York, 1981. zbl 0537.14012; MR0638599.
  • 20. G. Faltings, Endlichkeitss\``atze f\''ur abelsche Variet\``aten \''uber Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349-366. DOI 10.1007/BF01388432; zbl 0588.14026; MR0718935.
  • 21. J.-M. Fontaine, Représentations \(p\)-adiques semi-stables, Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque no. 223 (1994), 113-184. MR1293972.
  • 22. L. Gerritzen, Über Endomorphismen nichtarchimedischer holomorpher Tori, Invent. Math. 11 (1970), 27-36. DOI 10.1007/BF01389803; zbl 0199.55602; MR0286799.
  • 23. L. Gerritzen, On multiplication algebras of Riemann matrices, Math. Ann. 194 (1971), 109-122. DOI 10.1007/BF01362538; zbl 0245.14016; MR0288141.
  • 24. L. Gerritzen, M. van der Put, Schottky Groups and Mumford Curves, Springer (1980). DOI 10.1007/BFb0089957; zbl 0442.14009; MR0590243.
  • 25. M. Gros, Sur les \((K_0,\phi,N)\)-structures attachées aux courbes de Mumford, Rend. Sem. Mat. Univ. Padova 103 (2000), 233-260. zbl 1106.14303; MR1789541.
  • 26. A. Grothendieck, Mod\``eles de N\''eron et monodromie, Groupes de monodromie en géometrie algébrique, SGA 7 I, Lecture Notes in Math. 288, Springer (1972), 313-523. zbl 0248.14006.
  • 27. O. Hyodo, K. Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque No. 223 (1994), 221-268. zbl 0852.14004; MR1293974.
  • 28. L. Illusie, Ordinarité des intersections compl\``{e}tes g\''{e}nérales, The Grothendieck Festschrift, Vol. II, 376-405, Progr. Math., 87, Birkhäuser Boston 1990. zbl 0728.14021; MR1106904.
  • 29. L. Illusie, Réduction semi-stable ordinaire, cohomologie étale \(p\)-adique et cohomologie de de Rham d'apr\``es Bloch-Kato et Hyodo, appendix to B. Perrin-Riou, Repr\''esentations \(p\)-adiques ordinaires. ( Bures-sur-Yvette, 1988), Astérisque No. 223 (1994), 185-220. zbl 1043.11532; MR1293973.
  • 30. T. Ito, Weight-monodromy conjecture for \(p\)-adically uniformized varieties, Invent. Math. 159 (2005), no. 3, 607-656. DOI 10.1007/s00222-004-0395-y; zbl 1154.14014; MR2125735; arxiv math/0301201.
  • 31. U. Jannsen, Weights in arithmetic geometry, Jpn. J. Math. 5 (2010), no. 1, 73-102. DOI 10.1007/s11537-010-0947-4; zbl 1204.14011; MR2609323; arxiv 1003.0927.
  • 32. S. Kadziela, Rigid analytic uniformization of curves and the study of isogenies, Acta Appl. Math. 99 (2007), no. 2, 185-204. DOI 10.1007/s10440-007-9162-6; zbl 1140.14017; MR2350208.
  • 33. N. Katz, W. Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73-77. DOI 10.1007/BF01405203; zbl 0275.14011; MR0332791.
  • 34. W. Kim, K. Madapusi Pera, 2-adic integral canonical models, Forum Math. Sigma 4 (2016), e28, 34 pp. DOI 10.1017/fms.2016.23; zbl 1362.11059; MR3569319.
  • 35. M. Kuga, I. Satake, Abelian varieties attached to polarized \(K_3\)-surface, Math. Ann. 169 (1967), 239-242. DOI 10.1007/BF01399540; zbl 0221.14019; MR0210717.
  • 36. K. Künnemann, Projective regular models for abelian varieties, semistable reduction, and the height pairing, Duke Math. J. 95 (1998), no. 1, 161-212. DOI 10.1215/S0012-7094-98-09505-9; zbl 0955.14017; MR1646554.
  • 37. A. Kurihara, Construction of \(p\)-adic unit balls and the Hirzebruch proportionality, Am. J. Math. 102, 565-648 (1980). DOI 10.2307/2374116; zbl 0498.14011; MR0573103.
  • 38. C. Lazda, A. Pál, Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz (1,1) theorem, Compositio Math. 155 (2019), 1025-1045. DOI 10.1112/S0010437X19007164; zbl 07053890; MR3946283.
  • 39. B. Le Stum, La structure de Hyodo-Kato pour les courbes, Rend. Sem. Mat. Univ. Padova 94 (1995), 279-301. zbl 0874.14014; MR1370917.
  • 40. J. Lubin, J. Tate, Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. France 94 (1966), 49-59. DOI 10.24033/bsmf.1633; zbl 0156.04105; MR0238854.
  • 41. W. Lütkebohmert, Rigid geometry of curves and their Jacobians, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 61. Springer (2016). DOI 10.1007/978-3-319-27371-6; zbl 1387.14003; MR3467043.
  • 42. K. Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic, Invent. Math. 201 (2015), no. 2, 625-668. DOI 10.1007/s00222-014-0557-5; zbl 1329.14079; MR3370622; arxiv 1301.6326.
  • 43. T. Matsusaka, The criteria for algebraic equivalence and the torsion group, Amer. J. Math. 79 (1957), 53-66. DOI 10.2307/2372383; zbl 0077.34303; MR0082730.
  • 44. D. Maulik, Supersingular K3 surfaces for large primes, With an appendix by Andrew Snowden, Duke Math. J. 163 (2014), no. 13, 2357-2425. DOI 10.1215/00127094-2804783; zbl 1308.14043; MR3265555; arxiv 1203.2889.
  • 45. B. Mazur, J. Tate, J. Teitelbaum, On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), no. 1, 1-48. DOI 10.1007/BF01388731; zbl 0699.14028; MR0830037.
  • 46. F. Mokrane, La suite spectrale des poids en cohomologie de Hyodo-Kato, Duke Math. J. 72 (1993), no. 2, 301-337. DOI 10.1215/S0012-7094-93-07211-0; zbl 0834.14010; MR1248675.
  • 47. B. Moonen, On the Tate and Mumford-Tate conjectures in codimension \(1\) for varieties with \(h^{2,0}=1\), Duke Math. J. 166 (2017), no. 4, 739-799. DOI 10.1215/00127094-3774386; zbl 1372.14008; MR3619305; arxiv 1504.05406.
  • 48. D. Morrison, The Clemens-Schmid exact sequence and applications, Topics in transcendental algebraic geometry, 101-119, Ann. of Math. Stud. 106, Princeton Univ. Press (1984). zbl 0576.32034; MR0756848.
  • 49. M. Morrow, A Variational Tate Conjecture in crystalline cohomology. Preprint (2014). arxiv 1408.6783.
  • 50. D. Mumford, Abelian varieties, Oxford University Press (1970). zbl 0223.14022; MR0282985.
  • 51. D. Mumford, An analytic construction of degenerating abelian varieties over complete local rings, Compositio Math. 24 (1972), 239-272. zbl 0241.14020; MR0352106.
  • 52. G. A. Mustafin, Non-Archimedean uniformization, Mat. Sb. (N.S.) 105 (147) (1978), no. 2, 207-237, 287. zbl 0407.14006; MR0491696.
  • 53. Y. Nakkajima, \(p\)-adic weight spectral sequences of log varieties, J. Math. Sci. Univ. Tokyo 12 (2005), no. 4, 513-661. zbl 1108.14015; MR2206357.
  • 54. N. O. Nygaard, The Tate conjecture for ordinary K3 surfaces over finite fields, Invent. Math. 74 (1983), 213-237. DOI 10.1007/BF01394314; zbl 0557.14002; MR0723215.
  • 55. N. O. Nygaard, A. Ogus, Tate's conjecture for K3 surfaces of finite height, Ann. Math. 122 (1985), 461-507. DOI 10.2307/1971327; zbl 0591.14005; MR0819555.
  • 56. B. Perrin-Riou, Représentations \(p\)-adiques ordinaires. With an appendix by Luc Illusie. Périodes \(p\)-adiques (Bures-sur-Yvette, 1988), Astérisque no. 223 (1994), 185-220. zbl 1043.11532; MR1293973.
  • 57. U. Persson, On degenerations of algebraic surfaces, Mem. Amer. Math. Soc. 11, no. 189 (1977). DOI 10.1090/memo/0189; zbl 0368.14008; MR0466149.
  • 58. D. Petrequin, Classes de Chern et classes de cycles en cohomologie rigide, Bull. Soc. Math. France 131 (2003), no. 1, 59-121. DOI 10.24033/bsmf.2437; zbl 1083.14505; MR1975806; arxiv math/0103117.
  • 59. W. Raskind, A generalized Hodge-Tate conjecture for algebraic varieties with totally degenerate reduction over \(p\)-adic fields, Algebra and number theory, 99-115, Hindustan Book Agency, Delhi, 2005. zbl 1085.14009; MR2193347.
  • 60. W. Raskind, X. Xarles, On the étale cohomology of algebraic varieties with totally degenerate reduction over \(p\)-adic fields, J. Math. Sci. Univ. Tokyo 14 (2007), no. 2, 261-291. zbl 1138.14012; MR2351367; arxiv math/0306123.
  • 61. W. Raskind, X. Xarles, On \(p\)-adic intermediate Jacobians, Trans. Amer. Math. Soc. 359 (2007), no. 12, 6057-6077. DOI 10.1090/S0002-9947-07-04246-8; zbl 1127.14019; MR2336316; arxiv math/0601401.
  • 62. P. Schneider, U. Stuhler, The cohomology of \(p\)-adic symmetric spaces, Invent. Math. 105 (1991), no. 1, 47-122. DOI 10.1007/BF01232257; zbl 0751.14016; MR1109620.
  • 63. J.-P. Serre, Abelian \(\ell \)-Adic Representations and Elliptic Curves, W. A. Benjamin, Inc., New York-Amsterdam (1968). MR0263823.
  • 64. S. G. Tankeev, Algebraic cycles on surfaces and abelian varieties, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 2, 398-434, 463-464. zbl 0493.14014; MR0616226.
  • 65. S. G. Tankeev, Surfaces of type K3 over number fields and \(\ell \)-adic representations, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6, 1252-1271, 1328; translation in Math. USSR-Izv. 33 (1989), no. 3, 575-595. MR0984218.
  • 66. J. T. Tate, Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), 93-110, Harper and Row (1965). zbl 0213.22804; MR0225778.
  • 67. J. T. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134-144. DOI 10.1007/BF01404549; zbl 0147.20303; MR0206004.
  • 68. J. T. Tate, Conjectures on algebraic cycles in \(\ell \)-adic cohomology, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, 71-83, Amer. Math. Soc. (1994). zbl 0814.14009; MR1265523.
  • 69. B. Totaro, Recent progress on the Tate conjecture, Bull. Amer. Math. Soc. (N.S.) 54 (2017), no. 4, 575-590. DOI 10.1090/bull/1588; zbl 1375.14037; MR3683625.
  • 70. T. Tsuji, \(p\)-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), no. 2, 233-411. DOI 10.1007/s002220050330; zbl 0945.14008; MR1705837.
  • 71. N. Tsuzuki, Cohomological descent of rigid cohomology for proper coverings, Invent. Math. 151 (2003), no. 1, 101-133. DOI 10.1007/s00222-002-0252-9; zbl 1085.14019; MR1943743.
  • 72. G. Yamashita, The \(p\)-adic Lefschetz \((1,1)\) theorem in the semistable case, and the Picard number jumping locus, Math. Res. Lett. 18 (2011), no. 1, 107-124. DOI 10.4310/MRL.2011.v18.n1.a8; zbl 1238.14005; MR2770585.
  • 73. Ju. Zarhin, Endomorphisms of Abelian varieties over fields of finite characteristic, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 2, 272-277, 471. DOI 10.1070/IM1975v009n02ABEH001476; zbl 0345.14014; MR0371897.

Affiliation

Gregory, Oliver
TU M\``unchen, Zentrum Mathematik - M11, Boltzmannstr. 3, 85748 Garching bei M\''unchen, Germany
Liedtke, Christian
TU M\``unchen, Zentrum Mathematik - M11, Boltzmannstr. 3, 85748 Garching bei M\''unchen, Germany

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