We present a new approach for the proof of some formulas of Takuro Shintani concerning the Kronecker limit formulas for real quadratic fields.

In this paper, we study a discrete approximation to Brownian motion with varying dimension (BMVD) introduced by Chen and Lou in their 2019 paper with continuous time random walks on square lattices. The state space of BMVD contains a 2-dimensional component, a 3-dimensional component, and a “darning point” which joins these two components. Such a state space is equipped with the geodesic distance under which BMVD is a diffusion process. In this paper, we prove that BMVD restricted on a bounded domain containing the darning point is the weak limit of continuous-time reversible random walks with exponential holding times.

We prove that any symplectic automorphism of finite order on a manifold of type $\mathrm{OG}6$ acts trivially on the Beauville–Bogomolov–Fujiki lattice and that any birational transformation of finite order acts trivially on its discriminant group. Moreover, we classify all possible invariant and coinvariant sublattices.