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The algebraic K-theory of k[SL2(𝔽q)]

  • Isaac Moselle
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Abstract

We compute via trace methods the higher algebraic K-theory of the group ring k[SL2(𝔽q)], as well as the related groups PSL2(𝔽q), PGL2(𝔽q), and GL2(𝔽q), where k is a finite field of characteristic p and q=pr. At the core of the computation is the algebraic K-theory of the group ring of the Sylow p-subgroup, k[Cpr], which we determine via a theorem of LΓΌck--Reich--Rognes--Varisco on cyclic assembly for topological cyclic homology. In the process, we reprove the cyclic assembly result in the language of Nikolaus--Scholze, analyse assembly for smaller families of subgroups, and develop further tools for computing topological cyclic homology of group rings.

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Cite

BibTeXRIS
BibTeX
@misc{moselle2026algebraic,
  title = {{The algebraic K-theory of $k[\operatornameSL_2(\mathbbF_q)]$}},
  author = {Isaac Moselle},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.21421},
  eprint = {2606.21421},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.21421},
}
RIS
TY  - UNPB
TI  - The algebraic K-theory of k[SL_2(𝔽_q)]
AU  - Isaac Moselle
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2606.21421
UR  - https://arxiv.org/abs/2606.21421
AB  - We compute via trace methods the higher algebraic K-theory of the group ring k[SL_2(𝔽_q)], as well as the related groups PSL_2(𝔽_q), PGL_2(𝔽_q), and GL_2(𝔽_q), where k is a finite field of characteristic p and q=p^r. At the core of the computation is the algebraic K-theory of the group ring of the Sylow p-subgroup, k[C_p^r], which we determine via a theorem of LΓΌck--Reich--Rognes--Varisco on cyclic assembly for topological cyclic homology. In the process, we reprove the cyclic assembly result in the language of Nikolaus--Scholze, analyse assembly for smaller families of subgroups, and develop further tools for computing topological cyclic homology of group rings.
ER  -