Syntomic cohomology and Beilinson’s Tate conjecture for $K_2$
Authors:
Masanori Asakura and Kanetomo Sato
Journal:
J. Algebraic Geom. 22 (2013), 481-547
DOI:
https://doi.org/10.1090/S1056-3911-2012-00591-8
Published electronically:
October 3, 2012
MathSciNet review:
3048543
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Abstract |
References |
Additional Information
Abstract: We study Beilinson’s Tate conjecture for $K_2$ using the theory of syntomic cohomology. As an application, we construct integral indecomposable elements of $K_1$ of elliptic surfaces. Moreover, we give the first example of a surface $X$ with $p_g\ne 0$ over a $p$-adic field such that the torsion of $\mathrm {CH}_0(X)$ is finite.
References
- M. Artin and H. P. F. Swinnerton-Dyer, The Shafarevich-Tate conjecture for pencils of elliptic curves on $K3$ surfaces, Invent. Math. 20 (1973), 249–266. MR 417182, DOI https://doi.org/10.1007/BF01394097
- Masanori Asakura, Surjectivity of $p$-adic regulators on $K_2$ of Tate curves, Invent. Math. 165 (2006), no. 2, 267–324. MR 2231958, DOI https://doi.org/10.1007/s00222-005-0494-4
- Asakura, M.: On dlog image of $K_2$ of elliptic surface minus singular fibers. Preprint 2008, http://arxiv.org/abs/math/0511190
- Masanori Asakura and Shuji Saito, Surfaces over a $p$-adic field with infinite torsion in the Chow group of 0-cycles, Algebra Number Theory 1 (2007), no. 2, 163–181. MR 2361939, DOI https://doi.org/10.2140/ant.2007.1.163
- V. Srinivas (ed.), Cycles, motives and Shimura varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 21, Published for the Tata Institute of Fundamental Research, Mumbai; Narosa Publishing House, New Delhi, 2010. International distribution by American Mathematical Society. MR 2906018
- A. A. Beĭlinson, Higher regulators of modular curves, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 1–34. MR 862627, DOI https://doi.org/10.1090/conm/055.1/862627
- Spencer Bloch and Kazuya Kato, $p$-adic étale cohomology, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 107–152. MR 849653
- 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
- Christophe Breuil and William Messing, Torsion étale and crystalline cohomologies, Astérisque 279 (2002), 81–124. Cohomologies $p$-adiques et applications arithmétiques, II. MR 1922829
- Jean-Louis Colliot-Thélène and Wayne Raskind, ${\scr K}_2$-cohomology and the second Chow group, Math. Ann. 270 (1985), no. 2, 165–199. MR 771978, DOI https://doi.org/10.1007/BF01456181
- Jean-Louis Colliot-Thélène and Wayne Raskind, Groupe de Chow de codimension deux des variétés définies sur un corps de nombres: un théorème de finitude pour la torsion, Invent. Math. 105 (1991), no. 2, 221–245 (French, with English summary). MR 1115542, DOI https://doi.org/10.1007/BF01232266
- Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, and Christophe Soulé, Torsion dans le groupe de Chow de codimension deux, Duke Math. J. 50 (1983), no. 3, 763–801 (French). MR 714830, DOI https://doi.org/10.1215/S0012-7094-83-05038-X
- Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258
- P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 143–316. Lecture Notes in Math., Vol. 349 (French). MR 0337993
- Matthias Flach, A finiteness theorem for the symmetric square of an elliptic curve, Invent. Math. 109 (1992), no. 2, 307–327. MR 1172693, DOI https://doi.org/10.1007/BF01232029
- Jean-Marc Fontaine and William Messing, $p$-adic periods and $p$-adic étale cohomology, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 179–207. MR 902593, DOI https://doi.org/10.1090/conm/067/902593
- Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620
- Uwe Jannsen, Mixed motives and algebraic $K$-theory, Lecture Notes in Mathematics, vol. 1400, Springer-Verlag, Berlin, 1990. With appendices by S. Bloch and C. Schoen. MR 1043451
- Kazuya Kato, On $p$-adic vanishing cycles (application of ideas of Fontaine-Messing), Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 207–251. MR 946241, DOI https://doi.org/10.2969/aspm/01010207
- Kazuya Kato, A Hasse principle for two-dimensional global fields, J. Reine Angew. Math. 366 (1986), 142–183. With an appendix by Jean-Louis Colliot-Thélène. MR 833016, DOI https://doi.org/10.1515/crll.1986.366.142
- Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 1463703
- Kazuya Kato, The explicit reciprocity law and the cohomology of Fontaine-Messing, Bull. Soc. Math. France 119 (1991), no. 4, 397–441 (English, with French summary). MR 1136845
- Masato Kurihara, A note on $p$-adic étale cohomology, Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), no. 7, 275–278. MR 931263
- Andreas Langer, Selmer groups and torsion zero cycles on the selfproduct of a semistable elliptic curve, Doc. Math. 2 (1997), 47–59. MR 1443066
- Andreas Langer, $0$-cycles on the elliptic modular surface of level $4$, Tohoku Math. J. (2) 50 (1998), no. 2, 291–302. MR 1622078, DOI https://doi.org/10.2748/tmj/1178224979
- Andreas Langer and Shuji Saito, Torsion zero-cycles on the self-product of a modular elliptic curve, Duke Math. J. 85 (1996), no. 2, 315–357. MR 1417619, DOI https://doi.org/10.1215/S0012-7094-96-08514-2
- A. S. Merkur′ev and A. A. Suslin, $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Dokl. Akad. Nauk SSSR 264 (1982), no. 3, 555–559 (Russian). MR 659762
- Stephen J. M. Mildenhall, Cycles in a product of elliptic curves, and a group analogous to the class group, Duke Math. J. 67 (1992), no. 2, 387–406. MR 1177312, DOI https://doi.org/10.1215/S0012-7094-92-06715-9
- Nekovář, J.: Syntomic cohomology and $p$-adic regulators. Preprint, 1997
- Wieslawa Niziol, On the image of $p$-adic regulators, Invent. Math. 127 (1997), no. 2, 375–400. MR 1427624, DOI https://doi.org/10.1007/s002220050125
- Andreas Rosenschon and V. Srinivas, Algebraic cycles on products of elliptic curves over $p$-adic fields, Math. Ann. 339 (2007), no. 2, 241–249. MR 2324719, DOI https://doi.org/10.1007/s00208-007-0107-1
- Shuji Saito, On the cycle map for torsion algebraic cycles of codimension two, Invent. Math. 106 (1991), no. 3, 443–460. MR 1134479, DOI https://doi.org/10.1007/BF01243920
- Saito, S., Sato, K.: A $p$-adic regulator map and finiteness results for arithmetic schemes. Documenta Math. Extra Volume: Andrei A. Suslin’s Sixtieth Birthday, 525–594 (2010)
- Shuji Saito and Kanetomo Sato, A finiteness theorem for zero-cycles over $p$-adic fields, Ann. of Math. (2) 172 (2010), no. 3, 1593–1639. With an appendix by Uwe Jannsen. MR 2726095, DOI https://doi.org/10.4007/annals.2010.172.1593
- Sato, K.: Characteristic classes for $p$-adic étale Tate twists and the image of $p$-adic regulators. Preprint, http://front.math.ucdavis.edu/1004.1357
- Peter Schneider, Introduction to the Beĭlinson conjectures, Beĭlinson’s conjectures on special values of $L$-functions, Perspect. Math., vol. 4, Academic Press, Boston, MA, 1988, pp. 1–35. MR 944989
- Peter Schneider, $p$-adic points of motives, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 225–249. MR 1265555, DOI https://doi.org/10.1090/pspum/055.2/1265555
- Anthony J. Scholl, Integral elements in $K$-theory and products of modular curves, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 467–489. MR 1744957
- Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368
- Soulé, C.: Opérations en $K$-théorie algébrique. Canad. J. Math. 37, 488–550 (1985)
- Michael Spiess, On indecomposable elements of $K_1$ of a product of elliptic curves, $K$-Theory 17 (1999), no. 4, 363–383. MR 1706121, DOI https://doi.org/10.1023/A%3A1007739216643
- V. Srinivas, Algebraic $K$-theory, 2nd ed., Progress in Mathematics, vol. 90, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1382659
- Peter F. Stiller, The Picard numbers of elliptic surfaces with many symmetries, Pacific J. Math. 128 (1987), no. 1, 157–189. MR 883383
- John Tate, Relations between $K_{2}$ and Galois cohomology, Invent. Math. 36 (1976), 257–274. MR 429837, DOI https://doi.org/10.1007/BF01390012
- 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 https://doi.org/10.1007/s002220050330
- Takeshi Tsuji, On $p$-adic nearby cycles of log smooth families, Bull. Soc. Math. France 128 (2000), no. 4, 529–575 (English, with English and French summaries). MR 1815397
- Artin, M., Grothendieck, A., Verdier, J.-L., with Deligne, P., Saint-Donat, B.: Théorie des Topos et Cohomologie Étale des Schémas (Lecture Notes in Math. 269, 270, 305), Berlin, Springer, 1972–1973
References
- Artin, M., Swinnerton-Dyer, H. P. F.: The Shafarevich-Tate conjecture for pencils of elliptic curves on $K$3 surfaces. Invent. Math. 20, 249–266 (1973) MR 0417182 (54:5240)
- Asakura, M.: Surjectivity of $p$-adic regulators on $K_2$ of Tate curves. Invent. Math. 165, 267–324 (2006) MR 2231958 (2007d:19007)
- Asakura, M.: On dlog image of $K_2$ of elliptic surface minus singular fibers. Preprint 2008, http://arxiv.org/abs/math/0511190
- Asakura, M., Saito, S.: Surfaces over a $p$-adic field with infinite torsion in the Chow group of $0$-cycles. Algebra Number Theory 1, 163–181 (2007) MR 2361939 (2008k:14015)
- Asakura, M., Sato, K.: Beilinson’s Tate conjecture for $K_2$ of elliptic surface: survey and examples. In Cycles, Motives and Shimura Varieties (Ed. V. Srinivas), Tata Institute of Fundamental Research (2010), 35–58. MR 2906018 (2012i:14002)
- Beilinson, A.: Higher regulators of modular curves. In: Applications of Algebraic $K$-theory to Algebraic Geometry and Number theory (Contemp. Math. 55), pp. 1–34, Providence, Amer. Math. Soc., 1986 MR 862627 (88f:11060)
- Bloch, S., Kato, K.: $p$-adic étale cohomology. Inst. Hautes Études Sci. Publ. Math. 63, 107–152 (1986) MR 849653 (87k:14018)
- Bloch, S., Kato, K.: $L$-functions and Tamagawa numbers of motives. In: Cartier, P., Illusie, L., Katz, N. M., Laumon, G., Manin, Yu. I., Ribet, K. A. (eds.) The Grothendieck Festscherift I (Progr. Math. 86), pp. 333–400, Boston, Birkhäuser, 1990 MR 1086888 (92g:11063)
- Breuil, C., Messing, W.: Torsion étale and crystalline cohomologies. In: Cohomologies $p$-adiques et applications arithmétiques, II, (Astérisque No. 279), pp. 81–124, Marseille, Soc. Math. France 2002 MR 1922829 (2004k:14027)
- Colliot-Thélène, J.-L., Raskind, W.: $K_2$-cohomology and the second Chow group. Math. Ann. 270, 165–199 (1985) MR 771978 (86m:14005)
- Colliot-Thélène, J.-L., Raskind, W.: Groupe de Chow de codimension deux des variété sur un corps de numbres: Un théorème de finitude pour la torsion, Invent. Math. 105, 221–245 (1991) MR 1115542 (92j:14009)
- Colliot-Thélène, J.-L., Sansuc, J.-J., Soulé, C.: Torsion dans le groupe de Chow de codimension deux. Duke Math. J. 50, 763–801 (1983) MR 714830 (85d:14010)
- Deligne, P.: La conjecture de Weil I. Inst. Hautes Études Sci. Publ. Math. 43, 273–308 (1973) MR 0340258 (49:5013)
- Deligne, P., Rapoport, M.: Le schémas de modules de courbes elliptiques. In: Kuyk, W. (ed.), Modular functions of one variable II (Lecture Notes in Math. 349), pp. 143–316, Berlin, Springer, 1973 MR 0337993 (49:2762)
- Flach, M.: A finiteness theorem for the symmetric square of an elliptic curve. Invent. Math. 109, 307–327 (1992) MR 1172693 (93g:11066)
- Fontaine, J.-M., Messing, W.: $p$-adic periods and $p$-adic etale cohomology. In: Ribet, K. A. (ed.) Current Trends in Arithmetical Algebraic Geometry (Contemp. Math. 67), pp. 179–207, Providence, Amer. Math. Soc., 1987. MR 902593 (89g:14009)
- Hartshorne, R.: Local Cohomology (a seminar given by Grothendieck, A., Harvard University, Fall, 1961) (Lecture Notes in Math. 41), Berlin, Springer, 1967 MR 0224620 (37:219)
- Jannsen, U.: Mixed Motives and Algebraic $K$-theory (Lecture Notes in Math. 1400), Berlin, Springer, 1990 MR 1043451 (91g:14008)
- Kato, K.: On $p$-adic vanishing cycles (application of ideas of Fontaine-Messing). In: Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math. 10), pp. 207–251, Amsterdam, North-Holland, 1987. MR 946241 (89i:14014)
- Kato, K.: A Hasse principle for two-dimensional global fields (with an appendix by Colliot-Thélène, J.-L.), J. Reine Angew. Math. 366, 142–183 (1986) MR 833016 (88b:11036)
- Kato, K.: Logarithmic structures of Fontaine-Illusie. In: Igusa, J. (ed.) Algebraic Analysis, Geometry and Number Theory, pp. 191–224, Baltimore, The Johns Hopkins Univ. Press, 1988 MR 1463703 (99b:14020)
- Kato, K.: The explicit reciprocity law and the cohomology of Fontaine-Messing. Bull. Soc. Math. France 119, 397–441 (1991) MR 1136845 (92k:11068)
- Kurihara, M.: A note on $p$-adic étale cohomology. Proc. Japan Acad. Ser. A 63, 275–278 (1987) MR 931263 (89b:14026)
- Langer, A.: Selmer groups and torsion zero cycles on the self-product of a semistable elliptic curve. Doc. Math. 2, 47–59 (1997) MR 1443066 (98g:11074)
- Langer, A.: $0$-cycles on the elliptic modular surface of level $4$. Tohoku Math. J. 50, 315–360 (1998) MR 1622078 (99k:14009)
- Langer A., Saito, S.: Torsion zero-cycles on the self-product of a modular elliptic curve. Duke Math. J. 85, 315–357 (1996) MR 1417619 (97m:14005)
- Merkur$’$ev, A. S., Suslin, A. A.: $K$-cohomology of Severi-Brauer Varieties and the norm residue homomorphism, Math. USSR Izv. 21, 307–340 (1983) MR 659762 (84h:14025)
- Mildenhall, S.: Cycles in a product of elliptic curves, and a group analogous to the class group. Duke Math. J. 67, 387–406 (1992) MR 1177312 (93g:14014)
- Nekovář, J.: Syntomic cohomology and $p$-adic regulators. Preprint, 1997
- Niziol, W.: On the image of $p$-adic regulators. Invent. Math. 127, 375–400 (1997) MR 1427624 (98a:14031)
- Rosenschon, A., Srinivas, V.: Algebraic cycles on products of elliptic curves over $p$-adic fields. Math. Ann. 339, 241–249 (2007) MR 2324719 (2008j:14007)
- Saito, S.: On the cycle map for torsion algebraic cycles of codimension two. Invent. Math. 106, 443–460 (1991) MR 1134479 (93b:14011)
- Saito, S., Sato, K.: A $p$-adic regulator map and finiteness results for arithmetic schemes. Documenta Math. Extra Volume: Andrei A. Suslin’s Sixtieth Birthday, 525–594 (2010)
- Saito, S., Sato, K.: Finiteness theorem on zero-cycles over $p$-adic fields. With an appendix by Uwe Jannsen. Ann. of Math. (2) 172 (2010), no. 3, 1593–1639. MR 2726095 (2011m:14010)
- Sato, K.: Characteristic classes for $p$-adic étale Tate twists and the image of $p$-adic regulators. Preprint, http://front.math.ucdavis.edu/1004.1357
- Schneider, P.: Introduction to Beilinson’s conjecture. In: Rapoport, M., Schappacher, N., and Schneider, P. (eds.), Beilinson’s Conjectures on Special Values of $L$-Functions (Perspect. Math. 4), pp. 1–35, Boston, Academic Press, 1988. MR 944989 (89g:11053)
- Schneider, P.: $p$-adic point of motives. In: Jannsen, U. (ed.), Motives (Proc. Symp. Pure Math. 55-II), pp. 225–249, Providence, Amer. Math. Soc., 1994 MR 1265555 (95c:11077)
- Scholl, A. J.: Integral elements in $K$-theory and products of modular curves. In: Gordon, B. B., Lewis, J. D., Müller-Stach, S., Saito, S., Yui, N. (eds.) The arithmetic and geometry of algebraic cycles, Banff, 1998 (NATO Sci. Ser. C Math. Phys. Sci., 548), pp. 467–489, Dordrecht, Kluwer, 2000. MR 1744957 (2001i:11077)
- Silverman, J.: Advanced topics in the arithmetic of elliptic curves. Grad. Texts in Math. 15, New York, Springer 1994. MR 1312368 (96b:11074)
- Soulé, C.: Opérations en $K$-théorie algébrique. Canad. J. Math. 37, 488–550 (1985)
- Spiess, M.: On indecomposable elements of $K_1$ of a product of elliptic curves. $K$-Theory 17, 363–383 (1999) MR 1706121 (2000f:14013)
- Srinivas, V.: Algebraic $K$-Theory. 2nd ed. (Progr. Math. 90), Boston, Birkhäuser, 1996 MR 1382659 (97c:19001)
- Stiller, P.: The Picard numbers of elliptic surfaces with many symmetries. Pacific J. Math. 128, 157–189 (1987) MR 883383 (88c:14054)
- Tate, J.: Relations between $K_2$ and Galois cohomology. Invent. Math. 36, 257–274 (1976) MR 0429837 (55:2847)
- Tsuji, T.: $p$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137, 233–411 (1999) MR 1705837 (2000m:14024)
- Tsuji, T.: On $p$-adic nearby cycles of log smooth families. Bull. Soc. Math. France 128, 529–575 (2000) MR 1815397 (2002j:14024)
- Artin, M., Grothendieck, A., Verdier, J.-L., with Deligne, P., Saint-Donat, B.: Théorie des Topos et Cohomologie Étale des Schémas (Lecture Notes in Math. 269, 270, 305), Berlin, Springer, 1972–1973
Additional Information
Masanori Asakura
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
Email:
asakura@math.sci.hokudai.ac.jp
Kanetomo Sato
Affiliation:
Graduate school of Mathematics, Nagoya University, Nagoya 464-8602, Japan
Address at time of publication:
Department of Mathematics, Chuo University, Tokyo 112-8551, Japan
Email:
kanetomo@math.nagoya-u.ac.jp, kanetomo_sato@hotmail.com
Received by editor(s):
September 6, 2010
Received by editor(s) in revised form:
March 23, 2011
Published electronically:
October 3, 2012