Michelle Manes

Galois theory of quadratic rational functions

For a number field K with absolute Galois group G_K, we consider the action of G_K on the infinite tree of preimages of a point in K under a degree-two rational function phi, with particular attention to the case when phi commutes with a non-trivial Mobius transfomation. In a sense this is a dynamical systems analogue to the l-adic Galois representation attached to an elliptic curve, with particular attention to the CM case. Using a result about the discriminants of numerators of iterates of phi, we give a criterion for the image of the action to be as large as possible. This criterion is in terms of the arithmetic of the forward orbits of the two critical points of phi. In the case where phi commutes with a non-trivial Mobius transfomation, there is in effect only one critical orbit, and we give a modified version of our maximality criterion. We prove a Serre-type finite-index result in many cases of this latter setting.

Uniform bounds for preperiodic points in families of twists

Letbe a morphism of PN defined over a number field K. We prove that there is a bound B depending only onsuch that every twist of � has no more than B K-rational preperiodic points. (This result is analagous to a result of Silverman for abelian varieties (10).) For two specific families of quadratic rational maps over Q, we find the bound B explicitly.

A census of quadratic post-critically finite rational functions defined over

We find all quadratic post-critically finite (PCF) rational maps defined over the rationals. We describe an algorithm to search for possibly PCF maps. Using the algorithm, we eliminate all but twelve rational maps, all of which are verifiably PCF. We also give a complete description of possible rational preperiodic structures for quadratic PCF maps defined over Q.

Bad Reduction of Genus Three Curves with Complex Multiplication

Let C be a smooth, absolutely irreducible genus 3 curve over a number field M. Suppose that the Jacobian of C has complex multiplication by a sextic CM-field K. Suppose further that K contains no imaginary quadratic subfield. We give a bound on the primes \(\mathfrak{p}\) of M such that the stable reduction of C at \(\mathfrak{p}\) contains three irreducible components of genus 1.

Dynamical Belyi Maps

We study the dynamical properties of a large class of rational maps with exactly three ramification points. By constructing families of such maps, we obtain \({\mathcal O}(d^2)\) conservative maps of fixed degree d defined over \({\mathbb Q}\); this answers a question of Silverman. Rather precise results on the reduction of these maps yield strong information on their \({\mathbb Q}\)-dynamics.

Insufficiency of the Brauer–Manin Obstruction for Rational Points on Enriques Surfaces

In Varilly-Alvarado and Viray (Adv. Math. 226(6):4884–4901, 2011), the authors constructed an Enriques surface X over \(\mathbb{Q}\) with an etale-Brauer obstruction to the Hasse principle and no algebraic Brauer–Manin obstruction. In this paper, we show that the nontrivial Brauer class of \(X_{\overline{\mathbb{Q}}}\) does not descend to \(\mathbb{Q}\). Together with the results of Varilly-Alvarado and Viray (Adv. Math. 226(6):4884–4901, 2011), this proves that the Brauer–Manin obstruction is insufficient to explain all failures of the Hasse principle on Enriques surfaces.