Mathematische Zeitschrift 308, article 7 (2024). DOI: 10.1007/s00209-024-03566-w.
Preprint: arXiv:2307.16284
Abstract: Let K be a field, and let f ∈ K(z) be rational function. The preimages of a point x_0∈P^1(K) under iterates of f have a natural tree structure. As a result, the Galois group of the resulting field extensions of K naturally embed into the automorphism group of this tree. In unpublished work from 2013, Pink described a certain proper subgroup M_ℓ that this so-called arboreal Galois group G_∞ must lie in if f is quadratic and its two critical points collide at the ℓ-th iteration. After presenting a new description of M_ℓ and a new proof of Pink's theorem, we state and prove necessary and sufficient conditions for G_∞ to be the full group M_ℓ.
Research in Number Theory 8:45 (2022). DOI: 10.1007/s40993-022-00342-9.
Abstract: For each d ∈ N, we establish an infinite family of weight 1/2 quantum modular forms from the overpartition M_d-rank generating function. Infinite quantum families from both the Dyson rank overpartition generating function and the overpartition M_2-rank generating function appear as special cases of our work. As a corollary, we obtain explicit closed expressions which may be used to evaluate Eichler integrals of certain weight 3/2 theta functions.