Mahler Measures and Beilinson's Conjecture for Elliptic Curves over Real Quadratic Fields

By a formula of Villegas, the Mahler measures of $P_k(x,y)=x+1/x+y+1/y+k$ can be written as a Kronecker-Eisenstein series, where $k=\frac{4}{\sqrt{\lambda(2\tau)}}$ and $\lambda(\tau)=16\frac{\eta(\tau/2)^8\eta(2\tau)^{16}}{\eta(\tau)^{24}}$ is the modular lambda-function. If $\tau$ is taken to be CM points, we can use Villegas's formula to express the Mahler measures of $P_k$ as $L$-values of some modular forms. We prove that if $\tau_0$ is a CM point, then $\lambda(2\tau_0)$ is an algebraic number whose degree can be bounded by the product of the class numbers of $D_{\tau_0}$ and $D_{4\tau_0}$, where $D_{\tau}$ stands for the discriminant of a CM point $\tau$. Then we use the CM points with class numbers $\leqslant 2$ to obtain identities linking the Mahler measures of $P_k$ to $L$-values of modular forms. We get 35 identities, 5 of which are proven results, the others do not seem to appear in the literature. We also verify some cases of Beilinson's conjecture for elliptic curves over real quadratic fields that related to our discovery. Our method might also be used to other polynomial familys that mentioned in Villegas's paper.