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Shioda-Inose structure and elliptic K3 surfaces with high Mordell-Weil rank

Banff International Research Station for Mathematical Innovation and Discovery

If two $K3$ surfaces $X$ and $Y$ over $\mathbb{C}$ admit a rational map of finite degree $X\to Y$, Inose proved that their Picard numbers $\rho(X)$ and $\rho(Y)$ are equal. Suppose $X$ admits an elliptic fibration $\pi:X\to \mathbf{P}^{1}$. By a base change $b:\mathbf{P}^{1}\to \mathbf{P}^{1}$, we obtain another elliptic surface $\pi\times b:X':=X\times_{\mathbf{P}^{1}}\mathbf{P}^{1}\to \mathbf{P}^{1}$. If $X'$ is once again a $K3$ surface, we know $\rho(X')=\rho(X)$. However, it is difficult in general to find generators of the N\'eron-Severi goup of $X'$. Starting from various $K3$ surfaces $X$ having a Shioda-Inose structure, we construct $X'\to \mathbf{P}^{1}$ whose Mordell-Weil rank is large, and explore methods of finding generators of the Mordell-Weil group.

Mathematics; Algebraic geometry; Relativity and gravitational theory; Mathematical physics

video/mp4

Author affiliation: Chuo University

2018-07-23

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/66559

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/