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A generalized Abhyankar’s conjecture for simple Lie algebras in characteristic p>5

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Abstract

In the present paper, we study a purely inseparable counterpart of Abhyankar’s conjecture for the affine line in positive characteristic, and prove its validity for all the finite local non-abelian simple group schemes in characteristic \(p>5\). The crucial point is how to deal with finite local group schemes which cannot be realized as the Frobenius kernel of a smooth algebraic group. Such group schemes appear as the ones associated with Cartan type Lie algebras. We settle the problem for such Lie algebras by making use of natural gradations or filtrations on them.

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Acknowledgements

The main result in the present paper is a partial answer to the question given by Dmitriy Rumynin after the first named author’s talk at Max–Planck Institut für Mathematik in Bonn, September 2018. The authors would like to thank him for suggesting considering Cartan type Lie algebras as a test for the generalized Abhyankar’s conjecture. The first named author would like to thank Takuya Yamauchi for arranging the talk. The authors also thank them for giving helpful comments on the first draft of the present paper. The authors would like to thank Takao Yamazaki, Madhav Nori, Tomoyuki Abe and João Pedro dos Santos for having fruitful discussions and suggestions the authors received. Finally, the authors would like to thank the anonymous referee for his or her valuable comments which are very useful in improving the manuscript. The first named author was supported by JSPS Grant-in-Aid for JSPS Research Fellow, Grant number 19J00366. The second author was supported by GNSAGA of INdAM. The third author is supported by the Research Grants Council (RGC) of the Hong Kong SAR China (Project No. CUHK 14301019).

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Authors and Affiliations

  1. Department of Mathematics, School of Engineering, Tokyo Denki University, 5 Senju Asahi, Adachi, Tokyo, 120-8551, Japan

    Shusuke Otabe

  2. Dipartimento di Matematica e Informatica Ulisse Dini, Università degli Studi di Firenze, Viale Morgagni 67/a, Firenze, 50134, Italy

    Fabio Tonini

  3. Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

    Lei Zhang

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  1. Shusuke Otabe
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  2. Fabio Tonini
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  3. Lei Zhang
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Correspondence to Shusuke Otabe.

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Communicated by Vasudevan Srinivas.

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Otabe, S., Tonini, F. & Zhang, L. A generalized Abhyankar’s conjecture for simple Lie algebras in characteristic p>5. Math. Ann. 383, 1–54 (2022). https://doi.org/10.1007/s00208-021-02269-5

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