$L$-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings

We derive new integral presentations for central derivative values of $L$-functions of elliptic curves $E\mathrm{/}\mathbf{Q}$ twisted by ring class characters of a real quadratic field $K$ in terms of automorphic Green’s functions for certain Hirzebruch-Zagier-like arithmetic divisors on the product of modular curves $X_0(N)\times X_0(N)$ along real geodesic cycles. We also relate these sums to Birch-Swinnerton-Dyer constants and periods.