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Generic genus-one fibres of the integral I_r for qP

Establish that, for fixed s and t in the discrete Painlevé mapping (x,y,t) ↦ (t/(x−s^{-1}y), s x/y, s t), the algebraic curves defined by the level sets I_r(x,y)=c (where I_r is the integral of motion I_r = Tr[A(s^{r−1})⋯A(s)A(1)] − (t^r+1) with A(z) specified in the paper) have geometric genus generically equal to one, and prove that the genus drops to zero only when c satisfies c^4 + (−1)^r c^3 − 8 t^r c^2 − (−1)^r 36 t^r c + 16 t^{2r} − 27 t^r = 0.

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Background

Using a Lax pair (Murata, 2009), the authors construct explicit rational integrals of motion I_r for qP when s has finite multiplicative order r, leading to algebraic fibres I_r(x,y)=c that contain the orbits.

Computations suggest that these fibres are generically genus-one curves (i.e., elliptic-type) despite the non-autonomous nature of qP, with genus dropping to zero precisely on a closed subset determined by a specific polynomial condition in c and t. The conjecture formalizes this phenomenon.

References

In Section \ref{sec:genus}, we study the (geometric) genus of these curves, giving rise to a further conjecture that the genus must be generically one; see Conjecture \ref{conj:genus}. For fixed values of $s$ and $t$, the geometric genus of the fibres eq:fibres is generically one and reduces to zero only on the closed subset of ${c\in\mathbb{A}1}$ defined by \begin{equation}\label{eq:genuszerocondition} c4+(-1)r c3 - 8\, tr c2 - (-1)r 36\, tr c+ 16\, t{2r} - 27\, tr=0. \end{equation}

eq:genuszerocondition:

c4+(−1)rc3−8 trc2−(−1)r36 trc+16 t2r−27 tr=0.c^4+(-1)^r c^3 - 8\, t^r c^2 - (-1)^r 36\, t^r c+ 16\, t^{2r} - 27\, t^r=0.

eq:fibres:

{(x,y)∈Xt,s:Ir(x,y)=c},(c∈A1).\{(x,y)\in X_{t,s}:I_r(x,y)=c\},\qquad (c\in \mathbb{A}^1).

— Arithmetic dynamics of a discrete Painlevé equation (2508.18578 - Joshi et al., 26 Aug 2025) in Conjecture 3.4, Section 3.2 (Genera of fibres)