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PreprintCC BY-NC-SA 4.0arXiv

Rank stability of elliptic curves over prime cyclic extensions of function fields

  • Wanlin Li
  • Sun Woo Park
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Abstract

Let β„“ be an odd prime number. Let E be a non-isotrivial elliptic curve defined over the global function field K = 𝔽q(t) such that (q, 6) = 1 and q ≑ 1 mod β„“. We show that as q grows arbitrarily large, both analytic and algebraic ranks of E do not grow upon base change to 100% of β„€/β„“ β„€ Galois extensions over K. One of the applications we obtain is that over a positive proportion of β„€/β„“ β„€ Galois extensions L over K, the Tate--Shafarevich group of E over L is non-trivial. To obtain these results, we incorporate several geometric and probabilistic techniques which were studied in previous literature. These include the construction of desired auxiliary abelian varieties by Mazur--Rubin--Silverberg, the theory of middle convolution and big monodromy results by Katz and Hall, uniform Chebotarev density theorem by Kowalski, and distribution of prime Selmer groups of auxiliary abelian varieties by Klagsbrun--Mazur--Rubin and Park.

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Cite

BibTeXRIS
BibTeX
@misc{li2026rank,
  title = {{Rank stability of elliptic curves over prime cyclic extensions of function fields}},
  author = {Wanlin Li and Sun Woo Park},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03535},
  eprint = {2611.03535},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03535},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Rank stability of elliptic curves over prime cyclic extensions of function fields
AU  - Wanlin Li
AU  - Sun Woo Park
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03535
UR  - https://arxiv.org/abs/2611.03535
AB  - Let β„“ be an odd prime number. Let E be a non-isotrivial elliptic curve defined over the global function field K = 𝔽_q(t) such that (q, 6) = 1 and q ≑ 1 mod β„“. We show that as q grows arbitrarily large, both analytic and algebraic ranks of E do not grow upon base change to 100% of β„€/β„“ β„€ Galois extensions over K. One of the applications we obtain is that over a positive proportion of β„€/β„“ β„€ Galois extensions L over K, the Tate--Shafarevich group of E over L is non-trivial. To obtain these results, we incorporate several geometric and probabilistic techniques which were studied in previous literature. These include the construction of desired auxiliary abelian varieties by Mazur--Rubin--Silverberg, the theory of middle convolution and big monodromy results by Katz and Hall, uniform Chebotarev density theorem by Kowalski, and distribution of prime Selmer groups of auxiliary abelian varieties by Klagsbrun--Mazur--Rubin and Park.
ER  -