×

zbMATH — the first resource for mathematics

Cycles in a product of elliptic curves, and a group analogous to the class group. (English) Zbl 0788.14004
We consider a subgroup of the second Chow group of a surface, which we call \(\Sigma\). The group \(\Sigma\) is interesting because it is analogous to the divisor class group of classical number theory. It is defined as follows. Let \(X\) be a smooth algebraic surface defined over a number field \(K\) and suppose that \(X\) has a smooth proper model \({\mathcal X}\) defined over \({\mathfrak O}[1/N]\), a localization of the ring of integers in \(K\). Let \(j:X\to{\mathcal X}\) be the inclusion of \(X\) into \({\mathcal X}\). Then \(\Sigma\) is the kernel of the flat pull-back map \(j^*\): \(\Sigma:=\ker(j^*:\text{CH}^ 2({\mathcal X})\to\text{CH}^ 2(X))\). – In this paper we give examples of surfaces with finite \(\Sigma\), but where the Picard number of the special fibre is strictly greater than the Picard number of the generic fibre for infinitely many primes. To do this requires elements in \(H^ 1(X,{\mathcal K}_ 2)\) which do not come from \(K^ \times\bigotimes_ \mathbb{Z}\text{Pic}(X)\). We show \(\Sigma\) is torsion in other similar examples. – Our main theorem is the following one.
Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) with complex multiplication by the whole ring of integers in the quadratic imaginary number field \(K\). Let \(N\) be the discriminant of \(E\), and \({\mathcal E}\) be the smooth curve over \({\mathfrak O}[1/6N]\) defined by \(E\). Then the kernels \[ \Sigma_ K=\ker(\text{CH}^ 2({\mathcal E}\times{\mathcal E})\to\text{CH}^ 2(E_ K\times E_ K)) \] and \[ \Sigma=\ker(\text{CH}^ 2({\mathcal E}_{\mathbb{Z}[1/6N]}\times{\mathcal E}_{\mathbb{Z}[1/6N]})\to\text{CH}^ 2(E\times E)) \] are finite.
If \(E\) is defined over \(\mathbb{Q}\) and has a modular parametrization but no complex multiplication, then we can prove the following theorem: Let \(E\) be a modular elliptic curve defined over \(\mathbb{Q}\). Then \(\Sigma=\ker(j^*:\text{CH}^ 2{\mathcal X}_{\mathbb{Z}[1/5N]})\to\text{CH}^ 2(X))\) is torsion.
There are several conjectural reasons to expect \(\Sigma\) to be torsion or finite, which we now briefly discuss. Bass conjecture, that \(K_ 0({\mathcal Y})\) is finitely generated for \({\mathcal Y}\) a regular scheme of finite type over \(\mathbb{Z}\), together with the Grothendieck isomorphism \(K_ 0({\mathcal Y})\times\mathbb{Q}\simeq(\bigoplus_ i\text{CH}^ i({\mathcal Y}))\otimes\mathbb{Q}\), would imply that \(\Sigma\) has finite rank. If \(\text{CH}^ 2({\mathcal Y})\) is finitely generated, then \(\Sigma\) would be finitely generated. – An \(S\)-integral version of Beilinson’s conjectures would imply that \(\Sigma\) is torsion. – Finally, we mention that Bloch- Kato’s Tamagawa number conjectures imply that \(\Sigma\) is finite.

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
14G35 Modular and Shimura varieties
14H52 Elliptic curves
14C25 Algebraic cycles
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. Bloch, Algebraic \(K\)-theory, motives and algebraic cycles , to appear in Proceedings of 1990 ICM. · Zbl 0759.14001
[2] S. Bloch and K. Kato, \(L\)-functions and Tamagawa numbers of motives , The Grothendieck Festschrift, Vol. I ed. P. Cartier, et al., Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333-400. · Zbl 0768.14001
[3] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. · Zbl 0705.14001
[4] P. M. Cohn, Algebra. Vol. 2 , John Wiley & Sons, London-New York-Sydney, 1977. · Zbl 0341.00002
[5] J.-L. Colliot-Thélène and W. 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. · Zbl 0752.14004
[6] J.-L. Colliot-Thélène and J.-J. Sansuc, On the Chow groups of certain rational surfaces: a sequel to a paper of S. Bloch , Duke Math. J. 48 (1981), no. 2, 421-447. · Zbl 0479.14006
[7] 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, 143-316. Lecture Notes in Math., Vol. 349. · Zbl 0281.14010
[8] D. R. Dorman, Global orders in definite quaternion algebras as endomorphism rings for reduced CM elliptic curves , Théorie des nombres (Quebec, PQ, 1987) eds. J.-M. de Koninck, C. Levesque, and W. de Gruyter, de Gruyter, Berlin, 1989, pp. 108-116. · Zbl 0697.12011
[9] W. Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005
[10] T. Ibukiyama, On maximal orders of division quaternion algebras over the rational number field with certain optimal embeddings , Nagoya Math. J. 88 (1982), 181-195. · Zbl 0473.12012
[11] Kazuya Kato and Shuji Saito, Global class field theory of arithmetic schemes , 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. 255-331. · Zbl 0614.14001
[12] N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves , Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. · Zbl 0576.14026
[13] S. Lang, Introduction to Arakelov theory , Springer-Verlag, New York, 1988. · Zbl 0667.14001
[14] S. Lang, Elliptic functions , 2nd edition ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987. · Zbl 0615.14018
[15] B. Mazur, Modular curves and the Eisenstein ideal , Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33-186 (1978). · Zbl 0394.14008
[16] S. J. M. Mildenhall, Cycles in a Product of Curves , thesis, Univ. of Chicago.
[17] J. S. Milne, Jacobian varieties , Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 167-212. · Zbl 0604.14018
[18] A. P. Ogg, Hyperelliptic modular curves , Bull. Soc. Math. France 102 (1974), 449-462. · Zbl 0314.10018
[19] A. P. Ogg, Rational points on certain elliptic modular curves , Analytic number theory (Proc. Sympos. Pure Math., Vol XXIV, St. Louis Univ., St. Louis, Mo., 1972), Amer. Math. Soc., Providence, R.I., 1973, pp. 221-231. · Zbl 0273.14008
[20] A. P. Ogg, Modular functions , The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 521-532. · Zbl 0448.10021
[21] D. Ramakrishnan, Regulators, algebraic cycles, and values of \(L\)-functions , Algebraic \(K\)-theory and algebraic number theory (Honolulu, HI, 1987), Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 183-310. · Zbl 0694.14002
[22] W. 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. · Zbl 0669.14002
[23] W. Raskind, Torsion algebraic cycles on varieties over local fields , Algebraic \(K\)-theory: connections with geometry and topology (Lake Louise, AB, 1987) eds. J. F. Jardine and V. P. Snaith, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 279, Kluwer Acad. Publ., Dordrecht, 1989, pp. 343-388. · Zbl 0709.14005
[24] P. Salberger, Zero-cycles on rational surfaces over number fields , Invent. Math. 91 (1988), no. 3, 505-524. · Zbl 0688.14008
[25] J. H. Silverman, The arithmetic of elliptic curves , Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026
[26] W. C. Waterhouse, Abelian varieties over finite fields , Ann. Sci. École Norm. Sup. (4) 2 (1969), 521-560. · Zbl 0188.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.