An algebraic lifting invariant of Ellenberg, Venkatesh, and Westerland

Abstract

We define and prove basic properties of a lifting invariant of curves over an algebraically closed field k with a map to the projective line \({\mathbb {P}}^1_{k}\) that was introduced by Ellenberg, Venkatesh, and Westerland. In other work of Liu, Zureick-Brown, and the author, this lifting invariant, and its connection to components of Hurwitz schemes, is applied to help understand the average number of unramified extensions of function fields with a given Galois group.

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Acknowledgements

The author would like to thank Aaron Landesman, Yuan Liu, and the anonymous referee for comments on an earlier draft of this note, and Jordan Ellenberg, Akshay Venkatesh, and Craig Westerland for their support of the writing of this note. This work was done with the support of a Packard Fellowship for Science and Engineering, a Sloan Research Fellowship, and National Science Foundation grant DMS-1652116.

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Correspondence to Melanie Matchett Wood.

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Wood, M.M. An algebraic lifting invariant of Ellenberg, Venkatesh, and Westerland. Res Math Sci 8, 21 (2021). https://doi.org/10.1007/s40687-021-00259-2

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