Equidistribution for sets which are not necessarily Galois stable: on a theorem of Mignotte

On Clifford theory with Galois action

We study Galois actions on 2+1D topological quantum field theories (TQFTs), characterizing their interplay with theory factorization, gauging, the structure of gapped boundaries and dualities, 0-form symmetries, 1-form symmetries, and 2-groups. In order to gain a better physical understanding of Galois actions, we prove sufficient conditions for the preservation of unitarity. We then map out the Galois orbits of various classes of unitary TQFTs. The simplest such orbits are trivial (e.g., as in various theories of physical interest like the Toric Code, Double Semion, and 3-Fermion Model), and we refer to such theories as unitary “Galois fixed point TQFTs”. Starting from these fixed point theories, we study conditions for preservation of Galois invariance under gauging 0-form and 1-form symmetries (as well as under more general anyon condensation). Assuming a conjecture in the literature, we prove that all unitary Galois fixed point TQFTs can be engineered by gauging 0-form symmetries of theories built from Deligne products of certain abelian TQFTs.

Galois action on families of generalised Fermat curves

We present a quantitative version of Bilu's theorem on the limit distribution of Galois orbits of sequences of points of small height in the $N$-dimensional algebraic torus. Our result gives, for a given point, an explicit bound for the discrepancy between its Galois orbit and the uniform distribution on the compact subtorus, in terms of the height and the generalized degree of the point.

We prove a quantitative equidistribution statement for adelic homogeneous subsets whose stabilizer is maximal and semisimple. Fixing the ambient space, the statement is uniform in all parameters. We explain how this implies certain equidistribution theorems which, even in a qualitative form, are not accessible to measure-classification theorems. As another application, we describe another proof of property ( τ ) (\tau ) for arithmetic groups.

In this paper, we prove a quantitative equidistribution theorem for polynomial sequences in a nilmanifold, where the average is taken along spheres instead of cubes. To be more precise, let $\Omega \subseteq \mathbb {Z}^{d}$ be the preimage of a sphere $\mathbb {F}_{p}^{d}$ under the natural embedding from $\mathbb {Z}^{d}$ to $\mathbb {F}_{p}^{d}$ . We show that if a rational polynomial sequence $(g(n)\Gamma )_{n\in \Omega }$ is not equidistributed on a nilmanifold $G/\Gamma $ , then there exists a non-trivial horizontal character $\eta $ of $G/\Gamma $ such that $\eta \circ g \,\mod \mathbb {Z}$ vanishes on $\Omega $ .

Let K be an algebraically closed, complete, nonarchimedean field, let E/K be an elliptic curve, and let E denote the Berkovich analytic space associated to E/K. We study the μ-equidistribution of finite subsets of E(K), where μ is a certain canonical unit Borel measure on E. Our main result is an inequality bounding the error term when testing against a certain class of continuous functions on E. We then give two applications to elliptic curves over global function fields: We prove a function field analogue of the Szpiro-Ullmo-Zhang equidistribution theorem for small points, and a function field analogue of a result of Baker, Ih, and Rumely on the finiteness of S-integral torsion points. Both applications are given in explicit quantitative form.