Year: 2026
Type: preprint
Abstract: 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 $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(3)^{\oplus2}\oplus\mathcal{O}_{\mathbb{P}^1}(4)\oplus\mathcal{O}_{\mathbb{P}^1})$. 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 $\mathbb{C}$ 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.
Source: arXiv (Cornell University)
Author: Burek, Dominik
Institution: Jagiellonian University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215988626