Zassenhaus Filtrations and Golod-Shafarevich Methods for Ihara Constants

Abstract

Let X be a curve over 𝔽_q. Denote by N(X) and g(X) the number of 𝔽_q-rational points and the genus of X respectively. The Ihara constant A(q)=limsup_g(X)→∞N(X)/g(X) measures the asymptotic abundance of rational points on curves over 𝔽_q. It is a central problem in the study of algebraic geometry and has also attracted broader mathematical interest. In particular, the constant A(2) is discussed in the \emph{Optimization Constants in Mathematics} collection maintained by Terence Tao et al \cite[Number 33]{DIT26}. In this paper, we study the Zassenhaus filtration of pro-p groups arising from class field towers. By combining Golod--Shafarevich theorem, embedding problems and central class field theory, we establish a new criterion for the infinitude of class field towers. As an application, we obtain an improvement on the best known lower bounds of Ihara constant A(2) and A(3) which are explicitly given by A(2)≥ 0.317779⋯ and A(3)≥ 0.496074⋯.

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BibTeX RIS

BibTeX

@misc{ma2026zassenhaus,
  title = {{Zassenhaus Filtrations and Golod-Shafarevich Methods for Ihara Constants}},
  author = {Liming Ma and Chaoping Xing and Zugan Xing},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33649},
  eprint = {2610.33649},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33649},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - Zassenhaus Filtrations and Golod-Shafarevich Methods for Ihara Constants
AU  - Liming Ma
AU  - Chaoping Xing
AU  - Zugan Xing
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.33649
UR  - https://arxiv.org/abs/2610.33649
AB  - Let X be a curve over 𝔽_q. Denote by N(X) and g(X) the number of 𝔽_q-rational points and the genus of X respectively. The Ihara constant A(q)=limsup_g(X)→∞N(X)/g(X) measures the asymptotic abundance of rational points on curves over 𝔽_q. It is a central problem in the study of algebraic geometry and has also attracted broader mathematical interest. In particular, the constant A(2) is discussed in the \emph{Optimization Constants in Mathematics} collection maintained by Terence Tao et al \cite[Number 33]{DIT26}. In this paper, we study the Zassenhaus filtration of pro-p groups arising from class field towers. By combining Golod--Shafarevich theorem, embedding problems and central class field theory, we establish a new criterion for the infinitude of class field towers. As an application, we obtain an improvement on the best known lower bounds of Ihara constant A(2) and A(3) which are explicitly given by A(2)≥ 0.317779⋯ and A(3)≥ 0.496074⋯.
ER  -

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