Galois action on Fuchsian surface groups and their solenoids

Let C be a complex algebraic curve uniformized by a Fuchsian group Γ. In the first part of this paper we identify the automorphism group of the solenoid associated with Γ with the Belyaev completion of its commensurator {\mathrm{Comm}(\Gamma)} and we use this identification to show that the isomorphism class of this completion is an invariant of the natural Galois action of {\mathrm{Gal}(\mathbb{C}/\mathbb{Q})} on algebraic curves. In turn, this fact yields a proof of the Galois invariance of the arithmeticity of Γ independent of Kazhhdan’s. In the second part we focus on the case in which Γ is arithmetic. The list of further Galois invariants we find includes: (i) the periods of {\mathrm{Comm}(\Gamma)}, (ii) the solvability of the equations {X^{2}+\sin^{2}\frac{2\pi}{2k+1}} in the invariant quaternion algebra of Γ and (iii) the property of Γ being a congruence subgroup.

Galois conjugation and multiboundary entanglement entropy

We revisit certain natural algebraic transformations on the space of 3D topological quantum field theories (TQFTs) called “Galois conjugations.” Using a notion of multiboundary entanglement entropy (MEE) defined for TQFTs on compact 3-manifolds with disjoint boundaries, we give these abstract transformations additional physical meaning. In the process, we prove a theorem on the invariance of MEE along orbits of the Galois action in the case of arbitrary Abelian theories defined on any link complement in S3. We then give a generalization to non-Abelian TQFTs living on certain infinite classes of torus link complements. Along the way, we find an interplay between the modular data of non-Abelian TQFTs, the topology of the ambient spacetime, and the Galois action. These results are suggestive of a deeper connection between entanglement and fusion.

Galois action on homology of generalized Fermat Curves

The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups H. We provide a unified study of the action of both cover Galois group H and the absolute Galois group $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.

Fake Galois actions

We prove that for all non-abelian finite simple groups [Formula: see text], there exists a fake [Formula: see text]th Galois action on [Formula: see text] with respect to [Formula: see text], where [Formula: see text] is the universal covering group of [Formula: see text] and [Formula: see text] is any non-negative integer coprime to the order of [Formula: see text]. This is one of the two inductive conditions needed to prove an [Formula: see text]-modular analogue of the Glauberman–Isaacs correspondence.

Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic

We use explicit methods to study the [Formula: see text]-torsion points on the Jacobian variety of the Fermat quartic. With the aid of computer algebra systems, we explicitly give a basis of the group of [Formula: see text]-torsion points. We calculate the Galois action, and show that the image of the mod [Formula: see text] Galois representation is isomorphic to the dihedral group of order [Formula: see text]. As applications, we calculate the Mordell–Weil group of the Jacobian variety of the Fermat quartic over each subfield of the [Formula: see text]th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the [Formula: see text]th cyclotomic field. Thus, we complete Faddeev’s work in 1960.