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PreprintCC BY 4.0arXiv

A Smooth Shioda-Tate Isomorphism for Elliptically Fibered 4-Manifolds

  • Maria Morariu
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Abstract

This paper studies a subgroup of the mapping class group of an elliptically fibered $4$-manifold π: M→ B, inspired by number theory. The smooth Mordell-Weil group MW(π) is the group of isotopy classes of smooth sections of π. We construct a smooth analog of the Shioda-Tate isomorphism, identifying MW(π) with the quotient of H2(M;ℤ) by the trivial lattice of π. This shows that MW(π) can be computed explicitly from H2(M;ℤ). As an application, we show that the natural inclusion of the holomorphic Mordell-Weil group into the smooth one is an isomorphism if and only if M is a rational elliptic surface. As another application, we show that MW(π) satisfies a strong version of Nielsen realization: any torsion element of MW(π) is represented by a holomorphic translation of M of the same order.

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Cite

BibTeXRIS
BibTeX
@misc{morariu2026smooth,
  title = {{A Smooth Shioda-Tate Isomorphism for Elliptically Fibered 4-Manifolds}},
  author = {Maria Morariu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05177},
  eprint = {2611.05177},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05177},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Smooth Shioda-Tate Isomorphism for Elliptically Fibered 4-Manifolds
AU  - Maria Morariu
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05177
UR  - https://arxiv.org/abs/2611.05177
AB  - This paper studies a subgroup of the mapping class group of an elliptically fibered $4$-manifold π: M→ B, inspired by number theory. The smooth Mordell-Weil group MW(π) is the group of isotopy classes of smooth sections of π. We construct a smooth analog of the Shioda-Tate isomorphism, identifying MW(π) with the quotient of H^2(M;ℤ) by the trivial lattice of π. This shows that MW(π) can be computed explicitly from H^2(M;ℤ). As an application, we show that the natural inclusion of the holomorphic Mordell-Weil group into the smooth one is an isomorphism if and only if M is a rational elliptic surface. As another application, we show that MW(π) satisfies a strong version of Nielsen realization: any torsion element of MW(π) is represented by a holomorphic translation of M of the same order.
ER  -