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 -- P¹ with a list of n points q₁, , qₙ P¹ -- and partitions σ₁, , σₙ of d, how many degree d covers C P¹ are there with specified ramification σᵢ over qᵢ? We ask: for a generic source -- an r-pointed curve (C, p₁, , pᵣ) of genus 1 -- and partitions μ, σ₁, , σₙ of d with (μ) =r, how many degree d covers C P¹ are there with ramification profile μ over 0 corresponding to a fiber \p₁, , pᵣ\ and elsewhere ramification profiles σ₁, , σₙ? 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 σ₁ and σ₂ are ``even'' (consisting of 2, , 2), we go on to establish recursive formulas to compute invariants where each unmarked ramification profile is of the form (x, 1, , 1). A special case asks: given a generic d-pointed genus 1 curve (E, p₁, , pd), how many degree d covers (E, p₁, , pd) (P¹, 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.