Curves of genus two with maps of every degree to a fixed elliptic curve

TL;DR

The work classifies genus-2 curves $C$ over ${f C}$ mapping to a fixed elliptic curve $E$ with maps of every degree $n>1$. It combines a CM- and period-matrix–based analysis with a finite, computer-assisted search to enumerate all viable pairs $(C,E)$, showing there are exactly $20$ such pairs and associating each with a universal degree-quadratic form on a rank-4 map-module. Four distinct quaternary quadratic forms $q_1,q_2,q_3,q_4$ capture all degrees, and explicit models for the curves are produced via a CM- and Kani–Leprevost–Poonen framework, together with a detailed constructive description of the degree maps. The paper additionally proves universality results for the forms and provides explicit equations for the genus-2 curves and the corresponding degree-$n$ maps, enabling concrete geometric realizations and potentially informing reductions in positive characteristic. Overall, the work connects moduli of genus-2 curves with CM theory, explicit endomorphism data, and computational methods to achieve complete classification and realizations of these highly structured maps.

Abstract

We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from $C$ to $E$.

Figures (2)

• Figure 1: A strict fundamental domain ${\mathcal{F}}_1$ for $\Gamma(1)$
• Figure 2: A strict fundamental domain ${\mathcal{F}}_2$ for $\Gamma(2)$, whose closure is tiled with images of the closure of the strict fundamental domain ${\mathcal{F}}_1$. The tiles are labeled by the Möbius transformation that takes ${\mathcal{F}}_1$ to the given tile. Note that $(3+\sqrt{-3})/2$ and $(3+\sqrt{-3})/6$ are not included in ${\mathcal{F}}_2$.

Theorems & Definitions (22)

• Theorem 1
• Proposition 2
• Proposition 3
• Corollary 4
• Corollary 5
• Proposition 6
• Remark 7
• Remark 8
• proof : Proof of Proposition \ref{['P:structure']}
• Remark 9