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The Gross-Zagier formula on singular moduli for Shimura curves
Andrew Phillips
https://arxiv.org/abs/2509.11553 https://arxiv.org/pdf/2509.11553

The Gross-Zagier formula on singular moduli for Shimura curves
The Gross-Zagier formula on singular moduli can be seen as a calculation of the intersection multiplicity of two CM divisors on the integral model of a modular curve. We prove a generalization of this result to a Shimura curve.

Inverse curvature flow of closed Legendre curves
Takashi Kagaya, Masatomo Takahashi
https://arxiv.org/abs/2510.04566 https://arxiv.org/pdf/2510.04566

Inverse curvature flow of closed Legendre curves
In this paper, we deal with an inverse curvature flow of $\ell$-convex Legendre curves. Since the Legendre curve is a natural generalization of regular curve, the flow is a generalization of the classical inverse curvature flow of regular curves. For the initial value problem, we study on the unique existence of the flow in global time, the monotonicity of the number of the singular cusps with respect to t > 0 and the asymptotic behavior of the flow as $t \to \infty$. Regarding the asymptotics,…