A formula for the rank over ℚ(t) of the elliptic curve y²=x³+At⁶+Bt³+C

@misc{bao2026formula,
  title = {{A formula for the rank over $\mathbbQ(t)$ of the elliptic curve $y^2=x^3+At^6+Bt^3+C$}},
  author = {Zhengheng Bao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16349},
  eprint = {2609.16349},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16349},
}

TY  - UNPB
TI  - A formula for the rank over ℚ(t) of the elliptic curve y^2=x^3+At^6+Bt^3+C
AU  - Zhengheng Bao
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.16349
UR  - https://arxiv.org/abs/2609.16349
AB  - In this paper, we give an explicit formula for the rank and generators (up to finite index) over ℚ(t) of all non-trivial elliptic curves of the form y^2=x^3+At^6+Bt^3+C, which is a larger class of elliptic surfaces than the one in Kloosterman's paper (2026) and Desjardins and Naskrecki's paper (2024), namely y^2=x^3+At^6+C. Our proof provides a new method to find this formula using generators of the geometric Mordell-Weil group while this geometric Mordell-Weil group (which is a ℤ[ω]-module) no longer decomposes into rank-one submodules by the methods in Kloosterman's paper (2026) and Desjardins and Naskrecki's paper (2024). Moreover, our proof uses only a few elements in the Galois group of the coordinates of the generators and only light computations by hand.
ER  -