Hostname: page-component-669899f699-b58lm Total loading time: 0 Render date: 2025-04-28T19:32:26.188Z Has data issue: false hasContentIssue false
  • English
  • Français

On even K-groups over p-adic Lie extensions of global function fields

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  22 January 2025

Meng Fai Lim Open the ORCID record for Meng Fai Lim [Opens in a new window]
Meng Fai Lim*
Affiliation:
School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, P. R. China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let p be a fixed prime number, and let F be a global function field with characteristic not equal to p. In this article, we shall study the variation properties of the Sylow p-subgroups of the even K-groups in a p-adic Lie extension of F. When the p-adic Lie extension is assumed to contain the cyclotomic $\mathbb {Z}_p$-extension of F, we obtain growth estimate of these groups. We also establish a duality between the direct limit and inverse limit of the even K-groups.

Keywords

Global function fieldseven K-groupsp-adic Lie extensions

MSC classification

Primary: 11R23: Iwasawa theory 11R70: $K$-theory of global fields 11R34: Galois cohomology
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 15
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bass, H. and Tate, J., The Milnor ring of a global field . In: H. Bass (ed.), Algebraic K-theory, II: “Classical” algebraic $K$ -theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, WA), Lecture Notes in Mathematics, 342, Springer, Berlin, Heidelberg, 1972, pp. 349–446.Google Scholar
Bloch, S. and Kato, K., $p$ -adic étale cohomology . Inst. Hautes Études Sci. Publ. Math. No. 63(1986), 107–152.Google Scholar
Coates, J., On ${K}_2$ and some classical conjectures in algebraic number theory. Ann. Math. (2) 95(1972), 99–116.Google Scholar
Deng, L.-T., Kezuka, Y., Li, Y.-X., and Lim, M. F., Non-commutative Iwasawa theory of abelian varieties over global function fields. Preprint, arXiv:2405.20963 [math.NT].Google Scholar
Dixon, J., Du Sautoy, M. P. F., Mann, A., and Segal, D., Analytic pro- $p$ groups. 2nd ed., Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1999.Google Scholar
Dwyer, W. G. and Friedlander, E., Algebraic and etale $K$ -theory. Trans. Amer. Math. Soc. 292(1985), no. 1, 247–280.Google Scholar
Ferrero, B. and Washington, L. C., The Iwasawa invariant ${\mu}_p$ vanishes for abelian number fields. Ann. Math. (2) 109(1979), no. 2, 377–395.Google Scholar
Fukaya, T. and Kato, K., A formulation of conjectures on $p$ -adic zeta functions in noncommutative Iwasawa theory. In: Proceedings of the St. Petersburg mathematical society, Vol. 12, American Mathematical Society Translations: Series 2, Ser. 2, 219, American Mathematical Society, Providence, RI, 2006, pp. 1–85.Google Scholar
Geisser, T. and Levine, M., The $K$ -theory of fields in characteristic $p$ . Invent. Math. 139(2000), no. 3, 459–493.Google Scholar
Goodearl, K. R. and Warfield, R. B., An introduction to non-commutative Noetherian rings, London Mathematical Society Student Texts 61, Cambridge University Press, Cambridge, 2004.Google Scholar
Harder, G., Die Kohomologie $S$ -arithmetischer Gruppen über Funktionenkörpern. Invent. Math. 42(1977), 135–175.Google Scholar
Howson, S., Euler characteristics as invariants of Iwasawa modules . Proc. London Math. Soc. (3) 85(2002), no. 3, 634–658.Google Scholar
Howson, S., Structure of central torsion Iwasawa modules . Bull. Soc. Math. France 130(2002), no. 4, 507–535.Google Scholar
Iwasawa, K., On Zl-extensions of algebraic number fields . Ann. of Math. (2) 98(1973), 246–326.Google Scholar
Jannsen, U., Iwasawa modules up to isomorphism . In: Algebraic number theory, Advanced Studies in Pure Mathematics, 17, Academic Press, Boston, MA, 1989, pp. 171–207.Google Scholar
Ji, Q. and Qin, H., Iwasawa theory for ${K}_{2 nF}$ . J. K-Theory 12(2013), no. 1, 115–123.Google Scholar
Kolster, M., $K$ -theory and arithmetic. In: Contemporary developments in algebraic $K$ -theory, ICTP Lecture Notes Series, 15, The Abdus Salam International Centre for Theoretical Physics, Trieste, 2004, pp. 191–258.Google Scholar
Lai, K. F. and Tan, K.-S., A generalized Iwasawa’s theorem and its application . Res. Math. Sci. 8(2021), no. 2, Paper No. 20, 18 pp.Google Scholar
Lam, T. Y., Lectures on modules and rings, Graduate Texts in Mathematics, 189, Springer, New York, 1999.Google Scholar
Lei, A., Estimating class numbers over metabelian extensions . Acta Arith. 180(2017), 347–364.Google Scholar
Liang, D. and Lim, M. F., On the Iwasawa asymptotic class number formula for ${\mathbb{Z}}_p^r\rtimes {\mathbb{Z}}_p$ -extensions. Acta Arith. 189(2019), no. 2, 191–208.Google Scholar
Lim, M. F., A note on asymptotic class number upper bounds in $p$ -adic Lie extensions. Acta Math. Sin. (Engl. Ser.) 35(2019), no. 9, 1481–1490.Google Scholar
Lim, M. F., On the growth of even $K$ -groups of rings of integers in $p$ -adic Lie extensions. Israel J. Math. 249(2022), no. 2, 735–767.Google Scholar
Lim, M. F., Comparing direct limit and inverse limit of even $K$ -groups in multiple ${\mathbb{Z}}_p$ -extensions. J. Théor. Nr. Bordx. 36(2024) no. 2, pp. 537–555.Google Scholar
Lim, M. F. and Sharifi, R., Nekovář duality over $p$ -adic Lie extensions of global fields. Doc. Math. 18(2013), 621–678.Google Scholar
Milnor, J., Algebraic $K$ -theory and quadratic forms . Invent. Math. 9(1969/70), 318–344.Google Scholar
Nekovář, J., Selmer complexes . Astérisque 310(2006), viii+559 pp.Google Scholar
Neukirch, J., Schmidt, A., and Wingberg, K., Cohomology of number fields, 2nd ed., Grundlehren der mathematischen Wissenschaften, 323, Springer-Verlag, Berlin, 2008.Google Scholar
Neumann, A., Completed group algebras without zero divisors . Arch. Math. 51(1988), no. 6, 496–499.Google Scholar
Perbet, G., Sur les invariants d’Iwasawa dans les extensions de Lie $p$ -adiques . Algebra Number Theory 5(2011), 819–848.Google Scholar
Quillen, D., Higher algebraic $K$ -theory. I. In: Algebraic $K$ -theory, I: Higher $K$ -theories (Proc. Conf., Battelle Memorial Inst., Seattle, WA, 1972), Lecture Notes in Mathematics, 341, Springer, Berlin 1973, pp. 85–147.Google Scholar
Serre, J. -P., Sur la dimension cohomologique des groupes profinis . Topology 3(1965), pp. 413–420.Google Scholar
Serre, J.-P. and Algebra, L., Springer monographs in mathematics, Spring-Verlag, Berlin, 2000.Google Scholar
Soulé, C., $K$ -théorie des anneaux d’entiers de corps de nombres et cohomologie étale. Invent. Math. 55(1979), no. 3, 251–295.Google Scholar
Vauclair, D., Sur la dualité et la descente d’Iwasawa . Ann. Inst. Fourier Grenob. 59(2009), no. 2, 691–767.Google Scholar
Venjakob, O., On the structure theory of the Iwasawa algebra of a $p$ -adic Lie group. J. Eur. Math. Soc. 4(2002), no. 3, 271–311.Google Scholar
Voevodsky, V., On motivic cohomology with $\boldsymbol{Z}/l$ -coefficients. Ann. of Math. (2) 174(2011), no. 1, 401–438.Google Scholar
Weibel, C., The norm residue isomorphism theorem . J. Topol. 2(2009), no. 2, 346–372.Google Scholar
Weibel, C., The $K$ -book. An introduction to algebraic $K$ -theory, Graduate Studies in Mathematics, 145, American Mathematical Society, Providence, RI, 2013, xii+618 pp.Google Scholar
Witte, M., Non-commutative Iwasawa main conjecture . Int. J. Number Theory 16(2020), no. 9, 2041–2094.Google Scholar