Local height functions in the Arakelov theoretic approach to Quadratic Chabauty, I

In the Arakelov theoretic approach to the Quadratic Chabauty function on a curve $X $ is a $p $-adic log function on a trivial line bundle on $X $ with a non-trivial curvature. Viewed as a Vologodsky function it depends on the choice of the branch of the logarithm. The derivative with respect to the branch parameter then gives the local contributions to the Quadratic Chabauty function. Using the Katz-Litt Theorem we can give an explicit computation of this local contribution. In particular, we recover the description of the local contributions by Betts and Dogra. This is joint work with Mueller and Srinivasan. I will also sketch my new proof of the Katz-Litt Theorem, which follows similar lines.