Reverse Hurwitz counts of genus 1 curves

Abstract

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 $\ell(\mu)=r$, 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 $d$.