arXiv subjects
Publications and source records attributed to Gerold Schefer.
We prove a Galois equidistribution result for torsion points in $\mathbb G_m^n$ in the $p$-adic setting for test functions of the form $\log |F|_p$ where $F$ is a nonzero polynomial with coefficients in the $p$-adic numbers. Our result includes a power saving quantitative estimate of the decay rate rate of the equidistribution. As an application we show that Ih's Conjecture is true for a class of divisors of $\mathbb G_m^n$.
For every algebraic number $\kappa$ on the unit circle which is not a root of unity we prove the existence of a strict sequence of algebraic numbers whose height tends to zero, such that the averages of the evaluation of $f_\kappa(z)=\log|z -\kappa|$ in the conjugates are essentially bounded from above by $-h(\kappa)$. This completes a characterisation on functions $f_\kappa$ initiated by Autissier and Baker-Masser, who cover the cases $\kappa=2$ and $|\kappa|\ne 1$ respectively. Using the same ideas we also prove analogues in the $p$-adic setting.
We estimate the growth rate of the function which counts the number of torsion points of order at most $T$ on an algebraic subvariety of the algebraic torus $\mathbb G_m^n$ over some algebraically closed field. We prove a general upper bound which is sharp, and characterize the subvarieties for which the growth rate is maximal. For all other subvarieties there is a better bound which is power saving compared to the general one. Our result includes asymptotic formulas in characteristic zero where we use Laurent's Theorem, the Manin-Mumford Conjecture. However, we also obtain new upper bounds for $K$ the algebraic closure of a finite field.