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On the Inertia Conjecture and its generalizations

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Abstract

We study the two-point branched Galois covers of the projective line given by explicit equations and with prescribed branched data. We also obtain several results using formal patching techniques that are useful in realizing new Galois covers. As a consequence, we prove the Inertia Conjecture for the alternating groups Ap+1, Ap+3, Ap+4 when p ≡ 2 (mod 3) is an odd prime and for the group Ap+5 when additionally 4 ∤ (p + 1) and p ≥ 17. We also pose a general question motivated by the Inertia Conjecture and obtain some affirmative results. A special case of this question, which we call the Generalized Purely Wild Inertia Conjecture, is shown to be true for the groups for which the purely wild part of the Inertia Conjecture is already established. We show that if this generalized conjecture is true for the groups G1 and G2 which do not have a common quotient, then the conjecture is also true for the product G1 × G2.

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Acknowledgements

I would like to thank my PhD supervisor, Manish Kumar, for his suggestions and comments which helped to improve the structure and style of the manuscript. I am also indebted to the anonymous referee of this paper and the reviewers of my dissertation for their in-depth report and various suggestions to improve the presentation and writing style of the paper.

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Authors and Affiliations

  1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhaba Road, Colaba, Mumbai, 400005, India

    Soumyadip Das

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  1. Soumyadip Das
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Correspondence to Soumyadip Das.

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The author would like to thank Indian Statistical Institute, Bangalore Centre. This paper is part of the author’s PhD thesis.

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Das, S. On the Inertia Conjecture and its generalizations. Isr. J. Math. 253, 157–204 (2023). https://doi.org/10.1007/s11856-022-2359-6

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  • DOI: https://doi.org/10.1007/s11856-022-2359-6

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