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On the prime Selmer ranks of cyclic prime twist families of elliptic curves over global function fields
- Part of
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- Journal:
- Compositio Mathematica / Volume 161 / Issue 12 / December 2025
- Published online by Cambridge University Press:
- 08 May 2026, pp. 3277-3320
- Print publication:
- December 2025
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- Article
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Fix a prime number p. Let
$\mathbb{F}_q$ be a finite field of characteristic coprime to 2 and 3, and containing the primitive pth root of unity 
$\mu_p$. Based on the works by Swinnerton-Dyer and Klagsbrun, Mazur, and Rubin, we prove that the probability distribution of the sizes of prime Selmer groups over a family of cyclic prime twists of non-isotrivial elliptic curves over 
$\mathbb{F}_q(t)$ satisfying a number of mild constraints conforms to the distribution conjectured by Poonen and Rains with explicit error bounds. The key tools used in proving these results are the Riemann hypothesis over global function fields, the Erdös–Kac theorem, and the geometric ergodicity of Markov chains.
