Non-thin rank jumps for double elliptic K3 surfaces

Pub Date : 2024-06-28 DOI:10.1007/s00229-024-01554-2
Hector Pasten, Cecília Salgado
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Abstract

For an elliptic surface \(\pi :X\rightarrow \mathbb {P}^1\) defined over a number field K, a theorem of Silverman shows that for all but finitely many fibres above K-rational points, the resulting elliptic curve over K has Mordell-Weil rank at least as large as the rank of the group of sections of \(\pi \). When X is a K3 surface with two distinct elliptic fibrations, we show that the set of K-rational points of \(\mathbb {P}^1\) for which this rank inequality is strict, is not a thin set, under certain hypothesis on the fibrations. Our results provide one of the first cases of this phenomenon beyond that of rational elliptic surfaces.

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双椭圆 K3 曲面的非薄级跃迁
对于定义在数域 K 上的椭圆曲面 \(\pi :X\rightarrow \mathbb {P}^1\),西尔弗曼(Silverman)的一个定理表明,除了有限多个 K 有理点之上的纤维之外,K 上的椭圆曲线的莫德尔-韦尔阶(Mordell-Weil rank)至少与 \(\pi \)的截面群的阶一样大。当 X 是一个有两个不同椭圆纤分的 K3 曲面时,我们证明了在纤分的特定假设下,秩不等式严格的 \(\mathbb {P}^1\) 的 K 有理点集合不是一个薄集。我们的结果提供了这一现象在有理椭圆曲面之外的第一个案例。
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