Hostname: page-component-5db58dd55d-htx7c Total loading time: 0 Render date: 2026-06-05T13:08:10.253Z Has data issue: false hasContentIssue false
  • English
  • Français

THE INTEGRALITY OF ALGEBRAIC WITT VECTORS OVER IMAGINARY QUADRATIC FIELDS IS DECIDABLE

Part of: Algebraic number theory: global fields Computational number theory

Published online by Cambridge University Press:  20 April 2026

YASUHIRO ISHITSUKA  and
TAKEO URAMOTO
YASUHIRO ISHITSUKA
Affiliation:
Institute of Mathematics for Industry, Kyushu University , Japan e-mail: ishitsuka.yasuhiro.751@m.kyushu-u.ac.jp
TAKEO URAMOTO*
Affiliation:
Graduate School of Science and Engineering, Kagoshima University , Japan
*
e-mail: uramoto@sci.kagoshima-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper proves that the integrality of algebraic Witt vectors over imaginary quadratic fields is decidable; based on this, some related problems are also discussed.

Keywords

class field theoryalgebraic Witt vectorsintegralitydecidability

MSC classification

Primary: 11R37: Class field theory
Secondary: 11Y16: Algorithms; complexity

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 23
Check for updates
Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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

Footnotes

The first author is supported by JSPS KAKENHI Grant-number 21K13773; the second author is supported by JSPS KAKENHI Grant-number 22K03248. This work was also supported by the 2021 IMI Joint Use Research Program Short-term Joint Research ‘New interplays between algebraic language theory and classical class field theory’. This work was done during the time the second author was a member of Institute of Mathematics for Industry, Kyushu University, to whom he is grateful for their hospitality.

Communicated by Michael Coons

References

Borger, J., ‘The basic geometry of Witt vectors, I: The affine case’, Algebra Number Theory 5(2) (2011), 231–285.CrossRefGoogle Scholar
Borger, J., ‘The basic geometry of Witt vectors, II: Spaces’, Math. Ann. 351(2011) (2011), 877–933.CrossRefGoogle Scholar
Borger, J. and de Smit, B., ‘Galois theory and integral models of $\varLambda$ -rings’, Bull. Lond. Math. Soc. 40(3) (2008), 439–446.10.1112/blms/bdn024CrossRefGoogle Scholar
Borger, J. and de Smit, B., ‘Lambda actions of rings of integers’, Preprint, 2011, arXiv:1105.4662.Google Scholar
Borger, J. and de Smit, B., ‘Explicit class field theory and the algebraic geometry of $\varLambda$ -rings’, Preprint, 2018, arXiv:1809.02295.Google Scholar
Buium, A., ‘Geometry of p-jets’, Duke Math. J. 82(2) (1996), 349–367.10.1215/S0012-7094-96-08216-2CrossRefGoogle Scholar
Deligne, P. and Ribet, K., ‘Values of Abelian $L$ -functions at negative integers over totally real fields’, Invent. Math. 59 (1980), 227–286.CrossRefGoogle Scholar
Gurney, L., ‘Elliptic curves with complex multiplication and $\varLambda$ -rings’, PhD Thesis, Australian National University, 2015.Google Scholar
Ishitsuka, Y. and Uramoto, T., ‘On the integrality of algebraic Witt vectors over imaginary quadratic fields’, in: Symbolic Computation in Software Science (SCSS’21) Short and Work-in-progress Papers (Ed. T. Kutsia) (Johannes Kepler University Linz, Austria, 2021), 28–33.Google Scholar
Rotman, J., Galois Theory, 2nd edn (Springer-Verlag, New York, 1998).CrossRefGoogle Scholar
Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions (Princeton University Press, Princeton, NJ, 1971).Google Scholar
Silverman, J., The Arithmetic of Elliptic Curves (Springer-Verlag, New York, 1986).10.1007/978-1-4757-1920-8CrossRefGoogle Scholar
Uramoto, T., ‘Semi-galois categories II: an arithmetic analogue of Christol’s theorem’, J. Algebra 508(2018) (2018), 539–568.10.1016/j.jalgebra.2018.04.033CrossRefGoogle Scholar
Uramoto, T., ‘Semi-galois categories I: the classical Eilenberg variety theory’, J. Pure Appl. Algebra 229 (2025), Article No. 107863.10.1016/j.jpaa.2025.107863CrossRefGoogle Scholar
Uramoto, T., ‘Semi-galois categories III: Witt vectors by deformations of modular functions’, Res. Math. Sci. 12 (2025), Article No. 40.10.1007/s40687-025-00525-7CrossRefGoogle Scholar
Uramoto, T., ‘Semi-galois categories IV: a deformed reciprocity law for Siegel modular functions’, Math. Z. 310 (2025), Article No. 78.CrossRefGoogle Scholar
Yalkinoglu, B., ‘On arithmetic models and functoriality of Bost–Connes systems. With an appendix by Sergey Neshveyev’, Invent. Math. 191 (2013), 383–425.10.1007/s00222-012-0396-1CrossRefGoogle Scholar