论文标题
零熵的Borcherds Lattices和K3表面
Borcherds lattices and K3 surfaces of zero entropy
论文作者
论文摘要
让$ L $成为一个均匀的双曲线晶格，无限许多简单的$（-2）$ - 根。如果$ l $ a Borcherds lattice承认具有有限的内部产品的各向同性矢量，所有简单的$（-2）$ - 根。我们表明，仅当$ l $的熵为零时，或者仅当$ l $的所有对称保留某些各向异性向量时，我们才表明这种情况是这种情况。我们获得了Borcherds Lattices的完整分类，包括$ 194 $ lattices。反过来，这提供了与几乎可解决的对称组的等级$ \ ge 5 $的双曲线晶格的分类。最后，我们将这些一般结果应用于K3表面。我们获得了零熵和无限自动形态组的K3表面的Picard Lattices分类，其中包括$ 193 $ lattices。特别是我们表明，所有kummer表面，所有超高的K3表面和所有覆盖富集表面的K3表面（除一个例外）都接受了正熵的自动形态。
Let $L$ be an even, hyperbolic lattice with infinitely many simple $(-2)$-roots. We call $L$ a Borcherds lattice if it admits an isotropic vector with bounded inner product with all the simple $(-2)$-roots. We show that this is the case if and only if $L$ has zero entropy, or equivalently if and only if all symmetries of $L$ preserve some isotropic vector. We obtain a complete classification of Borcherds lattices, consisting of $194$ lattices. In turn this provides a classification of hyperbolic lattices of rank $\ge 5$ with virtually solvable symmetry group. Finally, we apply these general results to the case of K3 surfaces. We obtain a classification of Picard lattices of K3 surfaces of zero entropy and infinite automorphism group, consisting of $193$ lattices. In particular we show that all Kummer surfaces, all supersingular K3 surfaces and all K3 surfaces covering an Enriques surface (with one exception) admit an automorphism of positive entropy.
