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**On the classification of degree 1 elliptic threefolds with constant \(j\)-invariant.**
*(English)*
Zbl 1283.14014

This paper deals with the classification of a class of threefolds over \(\mathbb{C}\) admitting elliptic fibrations to \(\mathbb{P}^2\). More precisely, the threefolds considered here are the ones admitting a degree 1 elliptic fibration to \(\mathbb{P}^2\), that is, that are birational to a degree 6 hypersurface in the weighted projective space \(\mathbb{P}(2,3,1,1,1)\). Further it is also assumed that they are neither birational to a cone over an elliptic surface in \(\mathbb{P}(2,3,1,1)\) nor to a product \(\mathbb{P}^2 \times E\). For such threefolds all possible Mordell-Weil groups are determined. Further the ones with fibers having \(j\)-invariant 1728 the equation is determined for two possible cases of the Mordell-Weil group, and similar results are obtained for \(j\)-invariant \(0\) as well.

The main method is the following. Treating a given threefold as an elliptic surface \(S\) over the function field of \(\mathbb{P}^1\), the Mordell-Weil group is the subgroup of the Mordell-Weil group of \(S_{\overline{K(\mathbb{P}^1)}}\) fixed by the Galois action of \(\mathrm{Gal} \left(\overline{K(\mathbb{P}^1)}, K(\mathbb{P}^1) \right)\). By the classification of elliptic surfaces according to K. Oguiso and T. Shioda [Comment. Math. Univ. St. Pauli 40, No. 1, 83–99 (1991; Zbl 0757.14011)], there is a small list fo the possible Mordell-Weil groups of such surfaces, and then geometric considerations lead to the right subgroups. The analysis of the special cases of the \(j\)-invariants uses a Hodge theoretic description of the Mordell-Weil group.

The main method is the following. Treating a given threefold as an elliptic surface \(S\) over the function field of \(\mathbb{P}^1\), the Mordell-Weil group is the subgroup of the Mordell-Weil group of \(S_{\overline{K(\mathbb{P}^1)}}\) fixed by the Galois action of \(\mathrm{Gal} \left(\overline{K(\mathbb{P}^1)}, K(\mathbb{P}^1) \right)\). By the classification of elliptic surfaces according to K. Oguiso and T. Shioda [Comment. Math. Univ. St. Pauli 40, No. 1, 83–99 (1991; Zbl 0757.14011)], there is a small list fo the possible Mordell-Weil groups of such surfaces, and then geometric considerations lead to the right subgroups. The analysis of the special cases of the \(j\)-invariants uses a Hodge theoretic description of the Mordell-Weil group.

Reviewer: Zsolt Patakfalvi (Princeton)

### MSC:

14J30 | \(3\)-folds |

11G05 | Elliptic curves over global fields |

14J27 | Elliptic surfaces, elliptic or Calabi-Yau fibrations |

### Citations:

Zbl 0757.14011
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\textit{R. Kloosterman}, Ill. J. Math. 55, No. 3, 771--803 (2011; Zbl 1283.14014)

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