## 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.

### MSC:

 14J30 $$3$$-folds 11G05 Elliptic curves over global fields 14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations

### Keywords:

elliptic fibration; 3-fold; Mordell-Weil group

Zbl 0757.14011
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### References:

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