2017/02/07 by Spence, Haden
#FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1702.01950
In their 2015 paper, Mertens and Rolen prove that for a certain level 6 "almost holomorphic" modular function P, the degree of P(τ) over ℚ for quadratic τ is as large as expected, settling a conjecture of Bruinier and Ono. Analogously for level 1 modular functions f, we expect ℚ(f(τ)) to have similar degree to ℚ(j(τ)). In this paper, I show for a wide class of level 1 almost holomorphic modular functions that \dfrac1M[ℚ(j(τ)):ℚ]≤ [ℚ(f(τ)):ℚ]≤[ℚ(j(τ)):ℚ] for all quadratic τ and some constant M. This is proven using techniques of o-minimality, and hence can easily be made uniform; the constant M depends only upon the "degree" of f (in a certain well-defined sense).