三维Landau-Ginzburg势的Mahler测度
Mahler Measure of 3D Landau-Ginzburg Potentials
摘要：
我们用Eisenstein-Kronecker级数表示了$23$族Laurent多项式的Mahler测度.这些洛朗多项式作为Fano $3$ -折叠上的Landau-Ginzburg势出现，其中$16$定义了通用Picard秩$19$的$K3$超曲面，其余的是通用Picard秩$< 19$。我们将每个有理奇异模处的马勒测度与$3$上的$L$ -函数的某个权重- $3$新形式的值联系起来。此外，我们发现这些族的Mahler测度之间存在$10$奇异关系.
英文摘要：
We express the Mahler measures of $23$ families of Laurent polynomials in terms of Eisenstein-Kronecker series. These Laurent polynomials arise as Landau-Ginzburg potentials on Fano $3$-folds, $16$ of which define $K3$ hypersurfaces of generic Picard rank $19$, and the rest are of generic Picard rank $< 19$. We relate the Mahler measure at each rational singular moduli to the value at $3$ of the $L$-function of some weight-$3$ newform. Moreover, we find $10$ exotic relations among the Mahler measures of these families.
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