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PreprintCC BY 4.0arXiv

The groups SL2(𝔽13) and SL2(𝔽19) are Galois over β„š

  • Emir Eray Karabiyik
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Abstract

In this paper, we show that SL2(𝔽13) and SL2(𝔽19) are Galois groups of totally real extensions of β„š. For each of these primes β„“, we find a polynomial of degree β„“+1 whose splitting field has Galois group PSL2(𝔽ℓ). These fields satisfy BΓΆge's criterion and the associated central embedding problem has a proper solution with Galois group SL2(𝔽ℓ). One can choose the resulting fields totally real via a quadratic twist. The degree $14$ polynomial is found through a genus one Hurwitz family of degree 14 covers. The degree 20 polynomial is found using Yang's modular equations for the Shimura curve X6*(1).

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Cite

BibTeXRIS
BibTeX
@misc{karabiyik2026groups,
  title = {{The groups $\operatornameSL_2(\mathbbF_13)$ and $\operatornameSL_2(\mathbbF_19)$ are Galois over $\mathbbQ$}},
  author = {Emir Eray Karabiyik},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03946},
  eprint = {2609.03946},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03946},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The groups SL_2(𝔽_13) and SL_2(𝔽_19) are Galois over β„š
AU  - Emir Eray Karabiyik
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03946
UR  - https://arxiv.org/abs/2609.03946
AB  - In this paper, we show that SL_2(𝔽_13) and SL_2(𝔽_19) are Galois groups of totally real extensions of β„š. For each of these primes β„“, we find a polynomial of degree β„“+1 whose splitting field has Galois group PSL_2(𝔽_β„“). These fields satisfy BΓΆge's criterion and the associated central embedding problem has a proper solution with Galois group SL_2(𝔽_β„“). One can choose the resulting fields totally real via a quadratic twist. The degree $14$ polynomial is found through a genus one Hurwitz family of degree 14 covers. The degree 20 polynomial is found using Yang's modular equations for the Shimura curve X_6^*(1).
ER  -