Reverse Hurwitz counts of genus 1 curves

In this paper, we study a problem that is in a sense a reversal of the Hurwitz counting problem. The Hurwitz problem asks: for a generic target – ${\mathbb{P}}^1$ with a list of $n$ points $q_1,\dots ,q_n\in {\mathbb{P}}^1$ – and partitions ${\sigma }_1,\dots ,{\sigma }_n$ of $d$, how many degree $d$ covers $C\to {\mathbb{P}}^1$ are there with specified ramification ${\sigma }_i$ over $q_i$? We ask: for a generic source – an $r$-pointed curve $(C,p_1,\dots ,p_r)$ of genus $1$ – and partitions $\mu ,{\sigma }_1,\dots ,{\sigma }_n$ of $d$ with $\mathrm{\ell }(\mu )=r$, how many degree $d$ covers $C\to {\mathbb{P}}^1$ are there with ramification profile $\mu$ over $0$ corresponding to a fiber $\{p_1,\dots ,p_r\}$ and elsewhere ramification profiles ${\sigma }_1,\dots ,{\sigma }_n$? While the enumerative invariants we study bear a similarity to generalized Tevelev degrees, they are more difficult to express in closed form in general. Nonetheless, we establish key results: after proving a closed form result in the case where the only non-simple unmarked ramification profiles ${\sigma }_1$ and ${\sigma }_2$ are ``even’’ (consisting of $2,\dots ,2$), we go on to establish recursive formulas to compute invariants where each unmarked ramification profile is of the form $(x,1,\dots ,1)$. A special case asks: given a generic $d$-pointed genus $1$ curve $(E,p_1,\dots ,p_d)$, how many degree $d$ covers $(E,p_1,\dots ,p_d)\to ({\mathbb{P}}^1,0)$ are there with $d-2$ unspecified points of $E$ having ramification index $3$? We show that the answer is an explicit quartic in $d$.