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 ₁,,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.