Skip to content
DraconiticResearch
Search Sources API
182,549 studies31 sourcesUpdated 51 min ago
PreprintCC BY 4.0arXiv

Golod--Shafarevich for arithmetic surfaces

  • Timo Keller
  • Carlo Pagano
Open at sourcePDFDOICite

Abstract

We construct Golod--Shafarevich towers of arithmetic surfaces over the integers. In particular, we obtain an arithmetic surface with a section and infinite geometric etale fundamental group, answering a question raised by Bost and Charles about the existence of such surfaces. Taking generic fibres gives curves over the rationals with infinite geometric etale towers in which a rational point splits completely. This answers questions posed by Ihara and by Frey, Kani and Völklein. Taking the generic fibres of the tower, we also obtain sequence of curves over the rationals whose genera goes to infinity and whose logarithmic conductors grows no more than linearly in the genera, answering a question asked by Venkatesh at PCMI in 2022. The authors worked together on this problem since 2022 and developed a general strategy. They were able to bring it to fruit only after a collaboration with GPT5.6 Sol, Fable and Aletheia a Gemini-powered internal agent developed at Google DeepMind.

Related

  1. Images of Golod-Shafarevich algebras with small growth
    Laurent Bartholdi, Agata Smoktunowicz
    Quarterly Journal of MathematicsPeer-reviewedLicence not statedFull text20 citedAlgebra & number theoryHAL
  2. Zassenhaus Filtrations and Golod-Shafarevich Methods for Ihara Constants
    Liming Ma, Chaoping Xing, Zugan Xing
    arXivPreprintCC BY-NC-SA 4.0Full textAlgebra & number theoryarXiv
  3. On the arithmetic of Tate-Shafarevich groups via Kolyvagin's conjecture
    Chan-Ho Kim
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. Exponential Growth of Mean Multiplicities in Length Spectra of Semi-Arithmetic Surfaces of Arbitrary Arithmetic Dimension
    A. Zuevsky
    arXivPreprintCC0Full textAnalysis & PDEsarXiv
  5. Variation of Shafarevich-Tate groups in Hida families
    Enrico Da Ronche, Matteo Longo, Stefano Vigni +1
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{keller2026golodshafarevich,
  title = {{Golod--Shafarevich for arithmetic surfaces}},
  author = {Timo Keller and Carlo Pagano},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06461},
  eprint = {2610.06461},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06461},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Golod--Shafarevich for arithmetic surfaces
AU  - Timo Keller
AU  - Carlo Pagano
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06461
UR  - https://arxiv.org/abs/2610.06461
AB  - We construct Golod--Shafarevich towers of arithmetic surfaces over the integers. In particular, we obtain an arithmetic surface with a section and infinite geometric etale fundamental group, answering a question raised by Bost and Charles about the existence of such surfaces. Taking generic fibres gives curves over the rationals with infinite geometric etale towers in which a rational point splits completely. This answers questions posed by Ihara and by Frey, Kani and Völklein. Taking the generic fibres of the tower, we also obtain sequence of curves over the rationals whose genera goes to infinity and whose logarithmic conductors grows no more than linearly in the genera, answering a question asked by Venkatesh at PCMI in 2022. The authors worked together on this problem since 2022 and developed a general strategy. They were able to bring it to fruit only after a collaboration with GPT5.6 Sol, Fable and Aletheia a Gemini-powered internal agent developed at Google DeepMind.
ER  -