On 𝑝-adic intermediate Jacobians

Raskind, Wayne; Xarles, Xavier

doi:10.1090/S0002-9947-07-04246-8

On $p$-adic intermediate Jacobians
HTML articles powered by AMS MathViewer

by Wayne Raskind and Xavier Xarles PDF

Trans. Amer. Math. Soc. 359 (2007), 6057-6077 Request permission

Abstract:

For an algebraic variety $X$ of dimension $d$ with totally degenerate reduction over a $p$-adic field (definition recalled below) and an integer $i$ with $1\leq i\leq d$, we define a rigid analytic torus $J^i(X)$ together with an Abel-Jacobi mapping to it from the Chow group $CH^i(X)_{hom}$ of codimension $i$ algebraic cycles that are homologically equivalent to zero modulo rational equivalence. These tori are analogous to those defined by Griffiths using Hodge theory over $\bf {C}$. We compare and contrast the complex and $p$-adic theories. Finally, we examine a special case of a $p$-adic analogue of the Generalized Hodge Conjecture.

References

Amnon Besser, A generalization of Coleman’s $p$-adic integration theory, Invent. Math. 142 (2000), no. 2, 397–434. MR 1794067, DOI 10.1007/s002220000093
Spencer Bloch, Algebraic cycles and values of $L$-functions, J. Reine Angew. Math. 350 (1984), 94–108. MR 743535, DOI 10.1515/crll.1984.350.94
Spencer Bloch and Kazuya Kato, $p$-adic étale cohomology, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 107–152. MR 849653, DOI 10.1007/BF02831624
Spencer Bloch and Kazuya Kato, $L$-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333–400. MR 1086888
S. Bloch, H. Gillet, and C. Soulé, Algebraic cycles on degenerate fibers, Arithmetic geometry (Cortona, 1994) Sympos. Math., XXXVII, Cambridge Univ. Press, Cambridge, 1997, pp. 45–69. MR 1472491
Caterina Consani, Double complexes and Euler $L$-factors, Compositio Math. 111 (1998), no. 3, 323–358. MR 1617133, DOI 10.1023/A:1000362027455
A.J. de Jong, Smoothness, alterations and semi-stability, Publ. Math. I.H.E.S. 83 (1996) 52-96.
Pierre Deligne, Théorie de Hodge. I, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 425–430 (French). MR 0441965
Ehud de Shalit, The $p$-adic monodromy-weight conjecture for $p$-adically uniformized varieties, Compos. Math. 141 (2005), no. 1, 101–120. MR 2099771, DOI 10.1112/S0010437X04000594
Gerd Faltings, $p$-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), no. 1, 255–299. MR 924705, DOI 10.1090/S0894-0347-1988-0924705-1
Gerd Faltings, Crystalline cohomology and $p$-adic Galois-representations, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25–80. MR 1463696
F. Guillén and V. Navarro Aznar, Sur le théorème local des cycles invariants, Duke Math. J. 61 (1990), no. 1, 133–155 (French). MR 1068383, DOI 10.1215/S0012-7094-90-06107-1
Phillip A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460-495; ibid. (2) 90 (1969), 496–541. MR 0260733, DOI 10.2307/1970746
B. H. Gross and C. Schoen, The modified diagonal cycle on the triple product of a pointed curve, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 3, 649–679 (English, with English and French summaries). MR 1340948, DOI 10.5802/aif.1469
A. Grothendieck, Hodge’s general conjecture is false for trivial reasons, Topology 8 (1969), 299–303. MR 252404, DOI 10.1016/0040-9383(69)90016-0
Urs T. Hartl, Semi-stability and base change, Arch. Math. (Basel) 77 (2001), no. 3, 215–221. MR 1865862, DOI 10.1007/PL00000484
Lars Hesselholt and Ib Madsen, On the $K$-theory of local fields, Ann. of Math. (2) 158 (2003), no. 1, 1–113. MR 1998478, DOI 10.4007/annals.2003.158.1
Osamu Hyodo, A note on $p$-adic étale cohomology in the semistable reduction case, Invent. Math. 91 (1988), no. 3, 543–557. MR 928497, DOI 10.1007/BF01388786
Luc Illusie, Cohomologie de de Rham et cohomologie étale $p$-adique (d’après G. Faltings, J.-M. Fontaine et al.), Astérisque 189-190 (1990), Exp. No. 726, 325–374 (French). Séminaire Bourbaki, Vol. 1989/90. MR 1099881
Luc Illusie, Ordinarité des intersections complètes générales, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 376–405 (French). MR 1106904
L. Illusie, Réduction semi-stable ordinaire, cohomologie étale $p$-adique et cohomologie de de Rham, d’après Bloch-Kato et Hyodo, in Périodes p-adiques, Séminaire de Bures, 1988 (J.-M. Fontaine ed.), Astérisque 223 (1994), 209-220.
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
Klaus Künnemann, Projective regular models for abelian varieties, semistable reduction, and the height pairing, Duke Math. J. 95 (1998), no. 1, 161–212. MR 1646554, DOI 10.1215/S0012-7094-98-09505-9
Marc Levine, The indecomposable $K_3$ of fields, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 2, 255–344. MR 1005161, DOI 10.24033/asens.1585
David I. Lieberman, Higher Picard varieties, Amer. J. Math. 90 (1968), 1165–1199. MR 238857, DOI 10.2307/2373295
Yu. I. Manin, Three-dimensional hyperbolic geometry as $\infty$-adic Arakelov geometry, Invent. Math. 104 (1991), no. 2, 223–243. MR 1098608, DOI 10.1007/BF01245074
Yu. Manin and V. Drinfeld, Periods of $p$-adic Schottky groups, J. Reine Angew. Math. 262(263) (1973), 239–247. MR 396582, DOI 10.1142/9789812830517_{0}012
A.S. Merkur’ev and A. Suslin, The group $K_3$ of a field, Izv. Akad. Nauk. SSSR (1989), English translation in Mathematics of the USSR: Izvestia (1989).
David Mumford, An analytic construction of degenerating curves over complete local rings, Compositio Math. 24 (1972), 129–174. MR 352105
David Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24 (1972), 239–272. MR 352106
Jacob P. Murre, Un résultat en théorie des cycles algébriques de codimension deux, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 23, 981–984 (French, with English summary). MR 777590
G. A. Mustafin, Non-Archimedean uniformization, Mat. Sb. (N.S.) 105(147) (1978), no. 2, 207–237, 287 (Russian). MR 0491696
M. Rapoport and Th. Zink, Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), no. 1, 21–101 (German). MR 666636, DOI 10.1007/BF01394268
Wayne Raskind, Higher $l$-adic Abel-Jacobi mappings and filtrations on Chow groups, Duke Math. J. 78 (1995), no. 1, 33–57. MR 1328751, DOI 10.1215/S0012-7094-95-07803-X
W. Raskind, A generalized Hodge-Tate conjecture for varieties with totally degenerate reduction over $p$-adic fields, in Algebra and Number Theory : Proceedings of the Silver Jubilee Conference University of Hyderabad, edited by Rajat Tandon. New Delhi, Hindustan Book Agency.
W. Raskind, On the $p$-adic Tate conjecture for divisors on varieties with totally degenerate reduction, in preparation.
Wayne Raskind, Algebraic $K$-theory, étale cohomology and torsion algebraic cycles, Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 311–341. MR 991983, DOI 10.1090/conm/083/991983
Michel Raynaud, Variétés abéliennes et géométrie rigide, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 473–477. MR 0427326
Michel Raynaud, 1-motifs et monodromie géométrique, Astérisque 223 (1994), 295–319 (French). Périodes $p$-adiques (Bures-sur-Yvette, 1988). MR 1293976
J.-P. Serre, Abelian $\ell$-adic representations and elliptic curves, Benjamin 1967.
Joseph Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1975/76), no. 3, 229–257. MR 429885, DOI 10.1007/BF01403146
Takeshi Tsuji, $p$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), no. 2, 233–411. MR 1705837, DOI 10.1007/s002220050330
André Weil, Foundations of algebraic geometry, American Mathematical Society, Providence, R.I., 1962. MR 0144898
André Weil, On Picard varieties, Amer. J. Math. 74 (1952), 865–894. MR 50330, DOI 10.2307/2372230