The fixed-point proportion of Lattès maps

Abstract

How often does an element of the geometric iterated Galois group of a Lattès map fix a vertex of the tree? Let A/K be an abelian variety over a separably closed field of characteristic 0, G <= Aut(A,0) finite, and alpha an isogeny of degree at least 2 normalising G. The Lattès map on X=A/G has a preimage tree governed by alpha-power torsion; we determine the fixed-point proportion in every dimension. For zeta in G the weight is the reciprocal of the number of fixed points of zeta in alpha-power torsion (zero if infinite); equivalently, it is a product of p-adic absolute values of determinants on the alpha-active subspaces; for scalar alpha=m the proportion is (1/|G|) times the sum over 1 != zeta in G of the product over p dividing m of the p-adic absolute value of det(1 - zeta | H_1(A,Z)). We deduce positivity and vanishing criteria, prove the groups G_infinity are martingale groups, determine the limit law, and reduce groups with translations to the fixed-origin case. In dimension one we determine ten values for scalar multipliers and nontrivial G <= Aut(E,0), and for the eight quadratic Lattès classes the proportions are 1/8, 1/4 and 5/16. Bridy-Jones-Kelsey-Lodge exclude the Lattès case from their main theorem; their Theorem 2.5 periodic-point bound is attained, via reductions modulo split primes p = 1 (mod 4), for the map (x^2+a)/(x^2-a), a^2 = -1, and we settle the exceptional case of the Fariña-Asategui-Radi conjecture.