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

Technical Summary: Curves of Genus Two with Maps of Every Degree to a Fixed Elliptic Curve

Problem Statement
The paper addresses a classical question in the theory of hyperelliptic integrals, originally framed by Legendre in 1828: Does there exist a genus-2 curve $C$ over the complex numbers $\mathbb{C}$ and an elliptic curve $E$ over $\mathbb{C}$ such that for every integer $n > 1$, there exists a morphism of degree $n$ from $C$ to $E$? While genus-2 curves admitting maps to elliptic curves have been studied for nearly two centuries, the specific condition of admitting maps of every degree had not been resolved. The paper investigates whether such pairs $(C, E)$ exist and, if so, classifies them up to isomorphism.

Methodology
The proof strategy relies on the structural relationship between a genus-2 curve $C$ admitting a degree-2 map to an elliptic curve $E$ and a second elliptic curve $F$. The methodology proceeds through the following logical steps:

1. Reduction to Quadratic Forms:
   Using results from Kani and earlier work by Picard, the authors establish that if $C$ has a degree-2 map to $E$, there exists a unique elliptic curve $F$ and an isomorphism $\psi: E[2] \to F[2]$ such that the Jacobian of $C$ is isogenous to the quotient of $E \times F$ by the graph of $\psi$.
   The set of maps from $C$ to $E$ corresponds to a $\mathbb{Z}$-lattice in the vector space $\text{Hom}(E, \text{Jac } C)$. The degree of these maps defines a positive definite quadratic form on this lattice. For $C$ to have maps of every degree $n > 1$, this quadratic form must represent every integer $n > 1$.

2. Restriction of Endomorphism Rings:
   By analyzing the existence of maps of degrees 3 and 4, the authors derive strict constraints on the endomorphism rings of $E$ and $F$. Specifically, they prove that $E$ must have complex multiplication (CM) and that there must exist a cyclic isogeny between $E$ and $F$ of a specific small degree $p$. This reduces the infinite search space to a finite list of possible pairs of discriminants $(\Delta_E, \Delta_F)$.

3. Computational Enumeration:
   For the finite list of candidate discriminant pairs, the authors perform a computational search using the software Magma. They:

   • Construct the number fields containing the $j$-invariants of the relevant elliptic curves.
   • Determine the sets of possible degrees for endomorphisms of $E$ and homomorphisms from $E$ to $F$ that satisfy the 2-torsion compatibility conditions.
   • Compute the period matrices and the associated quadratic forms for the resulting genus-2 curves.
   • Verify which forms represent all integers from 2 to 31 (a necessary condition for representing all integers $n > 1$).

4. Verification of Quadratic Forms:
   The authors prove that the surviving quadratic forms are isomorphic to one of four specific quaternary forms ($q_1, q_2, q_3, q_4$). They then provide a separate number-theoretic proof (Proposition 2) demonstrating that each of these four forms indeed represents every integer greater than 1.

5. Non-Existence Results:
   To address the general case where a curve does not admit maps of every degree, the authors utilize Kani's "refined Humbert invariant," a ternary quadratic form associated with any genus-2 curve. They prove that for any such curve, there exists a specific integer $n$ (bounded by 59 in Theorem 3, and by a large constant $kN$ in Theorem 4) for which no minimal map of degree $n$ exists.

Key Results

• Theorem 1 (Classification): Up to isomorphism, there are exactly twenty pairs $(C, E)$ satisfying the condition that $C$ is a genus-2 curve, $E$ is an elliptic curve, and for every $n > 1$, there is a degree-$n$ map from $C$ to $E$.

  • These pairs are characterized by specific discriminants for the endomorphism rings of $E$ and $F$.
  • The associated quadratic forms on the mapping lattices fall into exactly four isomorphism classes ($q_1, q_2, q_3, q_4$).
  • Table 1 in the paper provides the explicit period matrix parameters ($\tau, \sigma$) and discriminants for all twenty pairs.
  • Table 2 provides explicit polynomial models $y^2 = f(x)$ for the curves $C$ corresponding to these pairs.

• Theorem 3 (Non-Existence Bound): For any genus-2 curve $C$, there exists an integer $n \in \{2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 19, 31, 59\}$ such that there is no minimal map of degree $n$ from $C$ to any elliptic curve.

• Theorem 4 (General Non-Existence): A stronger, albeit less precise, bound is provided involving a large integer $N$ (the product of many primes up to 1511), showing that for any genus-2 curve, a minimal map of degree $n$ fails to exist for some $n \in \{2, 3, 13, kN\}$.

Significance and Claims
The paper claims to resolve a problem that, while framed in modern terminology, could have been understood by 19th-century researchers like Legendre, Jacobi, and Picard. The primary significance lies in the surprising existence of such curves; the authors note that it is "perhaps surprisingly" that such curves exist at all.

The work provides a complete classification of these exceptional curves, identifying exactly twenty isomorphism classes. It further demonstrates that these twenty cases are the only instances where maps of every degree exist. For all other genus-2 curves, the paper establishes that there is a "gap" in the degrees of minimal maps to elliptic curves, with the maximum possible gap bounded by 59.

The paper also briefly notes that similar results hold for ordinary curves in positive characteristic via Serre-Tate lifting, but explicitly states that the situation for supersingular curves in positive characteristic has not been investigated and may yield different examples.

Historical Context
The introduction situates the work within a lineage of research spanning nearly 200 years, citing contributions from Legendre (1828), Jacobi, Kowalevski, Poincaré, Picard, and Humbert. The authors emphasize that their problem is a natural extension of the classical study of reducing hyperelliptic integrals to elliptic integrals, specifically asking if such a reduction is possible via transformations of every degree.