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Rank stability of elliptic curves over 𝔽q(t) in residue classes

  • Steve Fan
  • Sun Woo Park
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Abstract

Fix a prime β„“. Let K = 𝔽q(t) be a global function field such that gcd(q,6) = 1 and q ≑ 1 pmod β„“. Let E be a non-isotrivial elliptic curve over K. Given a fixed monic polynomial Q over 𝔽q, and assuming some mild conditions on E, we show that the rank of E does not change with respect to a positive proportion of β„€/β„“ β„€ extensions K(√[β„“]f)/K, as f varies over the set of monic polynomials over 𝔽q such that f ≑ A pmod Q for any given polynomial A. We obtain this result by combining analytic and probabilistic techniques to study the distribution of certain prime Selmer groups of auxiliary abelian varieties constructed from these polynomials f.

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Cite

BibTeXRIS
BibTeX
@misc{fan2026rank,
  title = {{Rank stability of elliptic curves over $\mathbbF_q(t)$ in residue classes}},
  author = {Steve Fan and Sun Woo Park},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33792},
  eprint = {2609.33792},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33792},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Rank stability of elliptic curves over 𝔽_q(t) in residue classes
AU  - Steve Fan
AU  - Sun Woo Park
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33792
UR  - https://arxiv.org/abs/2609.33792
AB  - Fix a prime β„“. Let K = 𝔽_q(t) be a global function field such that gcd(q,6) = 1 and q ≑ 1 pmod β„“. Let E be a non-isotrivial elliptic curve over K. Given a fixed monic polynomial Q over 𝔽_q, and assuming some mild conditions on E, we show that the rank of E does not change with respect to a positive proportion of β„€/β„“ β„€ extensions K(√[β„“]f)/K, as f varies over the set of monic polynomials over 𝔽_q such that f ≑ A pmod Q for any given polynomial A. We obtain this result by combining analytic and probabilistic techniques to study the distribution of certain prime Selmer groups of auxiliary abelian varieties constructed from these polynomials f.
ER  -