An elliptic Calabi-Yau threefold of Mordell-Weil rank twelve
Abstract
We construct a smooth projective Calabi-Yau threefold with a flat elliptic fibration of Mordell-Weil rank twelve and Hodge numbers (h1,1,h2,1)=(25,1). The construction starts from the Kummer surfaces associated with the products of a fixed elliptic curve and the level-four modular family. The threefold is birational to an anticanonical quartic in ℙ(𝒪ℙ1(3)⊕2⊕𝒪ℙ1(4)⊕𝒪ℙ1). We give twelve independent sections and construct a projective crepant resolution of the associated Weierstrass model. Varying the fixed elliptic curve gives a family of quartic resolutions that is locally complete at the original member. The resulting fourfold is birational over ℂ to a diagonal order-four quotient of a product of four elliptic curves with j-invariant $1728$, and every smooth projective model has Kodaira dimension zero. An algebraic correspondence identifies the third cohomology of the original quartic resolution with the tensor product of the first cohomology of the fixed elliptic curve and the transcendental cohomology of the modular K3 surface. The Galois representation on the latter factor is associated with a weight-three CM newform.
Related
- Mordell--Weil groups of quasi-elliptic surfaces in characteristic 3
- Calabi-Yau Threefold Singularities and their Universal Quilts
- Involution-equivariant topological recursion and mirror symmetry for the affine binary dihedral Calabi--Yau threefold
- No Shortcuts to Infinity: Weil-Petersson Distance in Calabi-Yau Complex Structure Moduli Space
- Calabi-Bernstein type theorems for entire solutions of certain nonlinear elliptic equations in the plane
BibTeX
@misc{burek2026elliptic,
title = {{An elliptic Calabi-Yau threefold of Mordell-Weil rank twelve}},
author = {Dominik Burek},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01276},
eprint = {2610.01276},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01276},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An elliptic Calabi-Yau threefold of Mordell-Weil rank twelve
AU - Dominik Burek
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.01276
UR - https://arxiv.org/abs/2610.01276
AB - We construct a smooth projective Calabi-Yau threefold with a flat elliptic fibration of Mordell-Weil rank twelve and Hodge numbers (h^1,1,h^2,1)=(25,1). The construction starts from the Kummer surfaces associated with the products of a fixed elliptic curve and the level-four modular family. The threefold is birational to an anticanonical quartic in ℙ(𝒪_ℙ^1(3)^⊕2⊕𝒪_ℙ^1(4)⊕𝒪_ℙ^1). We give twelve independent sections and construct a projective crepant resolution of the associated Weierstrass model. Varying the fixed elliptic curve gives a family of quartic resolutions that is locally complete at the original member. The resulting fourfold is birational over ℂ to a diagonal order-four quotient of a product of four elliptic curves with j-invariant $1728$, and every smooth projective model has Kodaira dimension zero. An algebraic correspondence identifies the third cohomology of the original quartic resolution with the tensor product of the first cohomology of the fixed elliptic curve and the transcendental cohomology of the modular K3 surface. The Galois representation on the latter factor is associated with a weight-three CM newform.
ER -