Braid groups, elliptic curves, and resolving the quartic

Abstract: We show that, up to a natural equivalence relation, the only non-trivial, non-identity holomorphic maps $$\mathrm{Conf}_n\mathbb{C}\to\mathrm{Conf}_m\mathbb{C}$$ between unordered configuration spaces, where $m\in{3,4}$, are the resolving quartic map $$R\colon\mathrm{Conf}_4\mathbb{C}\to\mathrm{Conf}_3\mathbb{C}$$, a map $$\Psi_3\colon\mathrm{Conf}_3\mathbb{C}\to\mathrm{Conf}_4\mathbb{C}$$ constructed from the inflection points of elliptic curves in a family, and $\Psi_3\circ R$. This completes the classification of holomorphic maps $$\mathrm{Conf}_n\mathbb{C}\to\mathrm{Conf}_m\mathbb{C}$$ for $m\leq n$, extending results of Lin, Chen and Salter, and partially resolves a conjecture of Farb. We also classify the holomorphic families of elliptic curves over $\mathrm{Conf}_n\mathbb{C}$. To do this we classify homomorphisms between braid groups with few strands and $\mathrm{PSL}_2\mathbb{Z}$, then apply powerful results from complex analysis and Teichm\"uller theory. Furthermore, we prove a conjecture of Castel about the equivalence classes of endomorphisms of the braid group with three strands.