Kiran Kedlaya: The tame Belyi theorem in positive characteristic
Time: Wed 2021-09-22 17.00 - 18.00
Lecturer: Kiran Kedlaya (UC San Diego)
Abstract: We study the positive-characteristic analogue of a celebrated theorem of Belyi, which we formulate in two forms. The “weak Belyi theorem” is that an algebraic curve over a field of characteristic 0 can be written as a three-point cover of the projective line if and only if it can be defined over some number field. The “strong Belyi theorem” adds that any curve over a given number field K admits a three-point cover defined over K.
In positive characteristic, Belyi's theorem does not carry over because there are many covers with very little ramification (like Artin–Schreier coverings). However, for p > 2, Saidi established the “weak tame Belyi theorem”: an algebraic curve over a field of characteristic p can be written as a tame three-point cover of the projective line if and only if it can be defined over some finite field. Using a probabilistic argument in the style of Poonen's Bertini theorem over finite fields, we upgrade this to the “strong tame Belyi theorem”: any curve over a given finite field admits a three-point cover defined over that field.
The case of characteristic 2 is subtler because it is not obvious that a curve in characteristic 2 admits even a single tame finite map to the projective line. This was recently established for curves over an algebraically closed field by Sugiyama–Yasuda using a mod-2 analogue of the Schwarzian derivative; based on this, Anbar–Tutdere established the weak tame Belyi theorem in characteristic 2. We modify the Sugiyama–Yasuda construction to work over a finite field; this allows to deduce the strong tame Belyi theorem in characteristic 2.
Joint work with Daniel Litt and Jakub Witaszek.
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