论文标题
MAHLER措施和$ L $ - 椭圆形曲线的值
Mahler measures and $L$-values of elliptic curves over real quadratic fields
论文作者
论文摘要
罗德里格斯·维勒加斯（Rodriguez Villegas）的著名公式表明，$ m（k）$ p_k（x，y）= x+1/x+y+y+1/y+k $可以写成Kronecker-Eisenstein系列。我们证明，Villegas公式的$ K $的学位可以由CM积分的班级数字界定。这一事实使我们能够系统地得出$ 28 $的新身份，将$ m（k）$链接到cusp表格的$ l $ - 价值。在贝林森（Beilinson）的猜想的指导下，我们还证明了$ 5 $公式，该公式表达了$ l $ l $ - 椭圆形曲线的$ l $ values，而实际二次字段上则是$ 2 $ 2 $ $ m（k）$的$ 2 \ times 2 $。这扩展了Guo（本文的第二作者），JI，Liu和Qin的最新作品，其中他们处理了$ K = 4 \ pm 4 \ sqrt {2} $的情况。
A famous formula of Rodriguez Villegas shows that the Mahler measures $m(k)$ of $P_k(x,y)=x+1/x+y+1/y+k$ can be written as a Kronecker-Eisenstein series. We prove that the degree of $k$ in Villegas' formula can be bounded by the class numbers of CM points. This fact allows us to systematically derive $28$ new identities linking $m(k)$ to $L$-values of cusp forms. Guided by Beilinson's conjecture, we also prove $5$ formulas that express $L$-values of CM elliptic curves over real quadratic fields to some $2\times 2$ determinants of $m(k)$. This extends a recent work of Guo (the second author of this paper), Ji, Liu, and Qin, in which they dealt with the cases when $k=4\pm 4\sqrt{2}$.
