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Apr 30, 2024, 11:13:06 AMApr 30

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Hi everyone,

Our speaker today is Renee Bell from Lehman College and her talk is titled: "How do points on plane curves generate fields? Let me count the ways." (abstract below). The talk is at 11am in PDL C-401 and as usual we will go to lunch with the speaker after the talk. We will plan to be back before the Rainwater seminar starts.

In their program on diophantine stability, Mazur and Rubin suggest studying a curve $C$ over $\mathbb{Q}$ by understanding the field extensions of generated by a single point of $C$; in particular, they ask to what extent the set of such field extensions determines the curve. A natural question in arithmetic statistics along these lines concerns the size of this set: for a smooth projective curve $C$ how many field extensions of $\mathbb{Q}$ — of given degree and bounded discriminant — arise from adjoining a point of $C$? Can we further count the number of such extensions with specified Galois group? Asymptotic lower bounds for these quantities have been found for elliptic curves by Lemke Oliver and Thorne, for hyperelliptic curves by Keyes, and for superelliptic curves by Beneish and Keyes. We discuss similar asymptotic lower bounds that hold for all smooth plane curves $C$, using tools such as geometry of numbers, Hilbert irreducibility, Newton polygons, and linear optimization.

Bianca Viray (she/her)

Craig McKibben and Sarah Merner Professor of Mathematics

University of Washington

Seattle, WA 98195

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