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