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Michelle Manes

Publications and source records attributed to Michelle Manes.

25 records · Page 2Linked to original sources
arXiv2012

Mahler measure of some singular K3-surfaces

We study the Mahler measure of the three-variable Laurent polynomial x + 1/x + y + 1/y + z + 1/z - k where k is a parameter. The zeros of this polynomial define (after desingularization) a family of K3-surfaces. In favorable cases, the K3-surface has Picard number 20, and the Mahler measure is related to its L-function. This was first studied by Marie-Jose Bertin. In this work, we prove several new formulas, extending the earlier work of Bertin.

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arXiv2012

Computing conjugating sets and automorphism groups of rational functions

Let phi and psi be endomorphisms of the projective line of degree at least 2, defined over a noetherian commutative ring R with unity. From a dynamical perspective, a significant question is to determine whether phi and psi are conjugate (or to answer the related question of whether a given map phi has a nontrivial automorphism). We show that the space of automorphisms of P^1 conjugating phi to psi is a finite subscheme of PGL(2) (respectively that the automorphism group of phi is a finite group scheme). We construct efficient algorithms for computing the set of conjugating maps (resp. the group of automorphisms) when R is a field. Each of our algorithms takes advantage of different dynamical structures, so context (e.g., field of definition and degree of the map) determines the preferred algorithm. We have implemented them in Sage when R is a finite field or the field of rational numbers, and we give running times for computing automorphism groups for hundreds of random endomorphisms of P^1. These examples demonstrate the superiority of these new algorithms over a naive approach using Groebner bases.

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arXiv2012

Uniform bounds for pre-periodic points in families of twists

Let phi be a morphism of projective N-space defined over a number field K. We prove that there is a bound B depending only on phi such that every twist of phi has no more than B K-rational preperiodic points. (This result is analagous to a result of Silverman for abelian varieties.) For two specific families of quadratic rational maps over Q, we find the bound B explicitly.

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arXiv2012

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.

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arXiv2009

Moduli spaces for families of rational maps on P^1

Let phi: P^1 --> P^1 be a rational map defined over a field K. We construct the moduli space M_d(N) parameterizing conjugacy classes of degree-d maps with a point of formal period N and present an algebraic proof that M_2(N) is geometrically irreducible for N>1. Restricting ourselves to maps phi of arbitrary degree d >= 2 such that the composition h^{-1} phi h = phi for some nontrivial h in PGL_2, we show that the moduli space parameterizing these maps with a point of formal period N is geometrically reducible for infinitely many N.

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arXiv2008

Uniform Bounds on Pre-Images under Quadratic Dynamical Systems

For any elements b,c of a number field K, let G(b,c) denote the backwards orbit of b under the map f_c: C-->C given by f_c(x)=x^2+c. We prove an upper bound on the number of elements of G(b,c) whose degree over K is at most some constant B. This bound depends only on b, [K:Q], and B, and is valid for all b outside an explicit finite set. We also show that, for any N>3 and any b in K outside a finite set, there are only finitely many pairs of complex numbers (y,c) for which [K(y,c):K]<2^(N-3) and the value of the N-th iterate of f_c(x) at x=y is b. Moreover, the bound 2^(N-3) in this result is optimal.

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