Dynamical modular curves for quadratic polynomial maps

Abstract

Motivated by the dynamical uniform boundedness conjecture of Morton and Silverman, specifically in the case of quadratic polynomials, we give a formal construction of a certain class of dynamical analogues of classical modular curves. The preperiodic points for a quadratic polynomial map may be endowed with the structure of a directed graph satisfying certain strict conditions; we call such a graph admissible. Given an admissible graph G, we construct a curve X1(G) whose points parametrize quadratic polynomial maps -which, up to equivalence, form a one-parameter family -together with a collection of marked preperiodic points that form a graph isomorphic to G. Building on work of Bousch and Morton, we show that these curves are irreducible in characteristic zero, and we give an application of irreducibility in the setting of number fields. We end with a discussion of the Galois theory associated to the preperiodic points of quadratic polynomials, including a certain Galois representation that arises naturally in this setting.

Current trends and open problems in arithmetic dynamics

“…, n r ) that classify maps with marked points of several indicated periods, and even more generally, curves X dyn 1 (Γ) that classify maps with marked points and/or cycles with orbits having a specified graph structure. See [58,61] for further details.…”

Moduli Spaces for Dynamical Systems with Portraits

“…The classical example is the dynatomic curve X dyn 1 (n) whose points classify pairs (c, α) such that α is a point of exact (or formal) period n for the map x 2 + c, or more generally x d + c. (See Example 13.1 for further information about dynatomic curves and their description as portrait moduli spaces.) The following papers are among those that investigate X dyn 1 (n): Bousch [5], Buff-Epstein-Koch [6], Buff-Lei [7], Douady-Hubbard [13,14], Doyle [16], Doyle et al [17], Doyle-Poonen [18], Gao [24], Gao-Ou [25], Krumm [34,35], Lau-Schleicher [36], Morton [48]. These papers study topics such as smoothness, irreducibility, genus, and gonality of X dyn 1 (n), as well as reduction mod p and specialization properties.…”

“…This, in turn, can be used to give a simpler description of M N d [P], i.e., a description using fewer equations in a lower-dimensional ambient space. This is the approach taken by the first author in [16], where dynamical modular curves are constructed for polynomials x 2 + c with portraits P specified by generating sets. However, there are subtleties in this approach, since disconnectedness of P may lead to extraneous components in the naively constructed moduli space.…”

“…Then as an A-module, the ring R ′ is generated by the elements of S, and the relations among those elements are exactly those described by (16).…”

“…The kernel of this map is generated by the relations (16), where there is one relation for each choice of (i, j, h). Hence the kernel is generated by the row-span of M, i.e., by A N 2 n M. This gives the isomorphism…”