TY - ELEC
AU - Kuwata, Masato
PY - 2018
TI - Shioda-Inose structure and elliptic K3 surfaces with high Mordell-Weil rank
LA - eng
M3 - Moving Image
AB - 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.
N2 - 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.
UR - https://open.library.ubc.ca/collections/48630/items/1.0369010
ER - End of Reference