Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zero-cycles

This paper constructs a large family of pairwise non-isomorphic hyperelliptic curves mapping birationally into abelian surfaces isogenous to products of elliptic curves to generate rational equivalences in the Chow group of zero-cycles, thereby providing progress toward Beilinson's conjecture on the vanishing of the kernel of the Albanese map.

The Gist

Here is an explanation of the paper "Hyperelliptic Curves Mapping to Abelian Varieties and Applications to Beilinson's Conjecture for Zero-Cycles," translated into simple language with creative analogies.

The Big Picture: A Mathematical Mystery

Imagine you are trying to solve a giant puzzle about the shape of numbers and geometry. The puzzle is called Beilinson's Conjecture.

Think of a smooth, perfect geometric shape (like a sphere or a donut) defined by equations using only rational numbers (fractions like 1/2, 3/4, etc.). Mathematicians want to know: How many distinct "points" can you find on this shape using only rational numbers?

There is a specific rule in this puzzle called the Albanese Map. Think of this map as a "fingerprint scanner." It takes a collection of points on your shape and tries to identify them.

• The Conjecture says: If you use only rational numbers, this fingerprint scanner should be perfect. It should never confuse two different collections of points. In other words, if the scanner says two collections look the same, they must be the same.
• The Problem: For complex shapes (like a donut with many holes), this is incredibly hard to prove. In the world of complex numbers (which includes decimals and roots), the scanner is often broken; it confuses many different things. But the conjecture claims that in the strict world of rational numbers, the scanner works perfectly.

The Authors' Strategy: The "Hyperelliptic" Detour

The authors, Evangelia Gazaki and Jonathan Love, decided to tackle this problem for a specific type of shape called an Abelian Surface (think of it as a 2-dimensional donut, or a "double donut").

To prove the fingerprint scanner works, they needed to find a way to show that certain confusing collections of points are actually identical. They did this by building a bridge between two different worlds:

1. The Rational World: The strict world of fractions.
2. The Curve World: A world of special, wiggly lines called Hyperelliptic Curves.

The Analogy:
Imagine you have a messy room (the Abelian Surface) where you can't tell if two piles of toys are the same.

• The authors say: "Let's build a special slide (a Hyperelliptic Curve) that connects to the room."
• They slide toys down this slide. Because the slide has a special symmetry (it folds in half perfectly), any toy that lands on the slide creates a "perfect pair."
• If they can show that enough different slides exist, and that every toy in the room can be reached by sliding down one of them, they can prove that the messy piles are actually just perfect pairs. If everything is a perfect pair, the "fingerprint scanner" (the Albanese map) works perfectly!

How They Built the Slides (The Math Part)

The paper has two main parts:

Part 1: The Theory (The Blueprint)

They proved a theorem that says: If you can find enough of these special slides, the puzzle is solved.

• They showed that if you have a point on your shape, and you can find a slide that passes near it (or a multiple of it), then that point behaves nicely.
• They proved that if you have a "double donut" made from two smaller donuts (elliptic curves), you can build an infinite number of these slides.
• Key Insight: They realized that these slides correspond to rational curves (straight lines) on a related shape called a Kummer Surface. It's like looking at the shadow of the donut; the shadow has straight lines, and by "unfolding" the shadow back into the donut, you get your wiggly slides.

Part 2: The Construction (Building the Slides)

They didn't just say the slides exist; they showed you how to build them.

• They used a technique called an Elliptic Fibration. Imagine the Kummer surface is a stack of pancakes (elliptic curves).
• They found a way to draw lines across this stack of pancakes.
• By pulling these lines back down to the original "double donut," they created the Hyperelliptic curves.
• The Result: They found that for almost any size of "slide" (genus), they could build an infinite number of unique slides. This gave them a massive toolkit to test the conjecture.

The Real-World Test (Computations)

The authors didn't stop at theory. They wrote computer code to test this on real examples.

• They picked pairs of elliptic curves (the "ingredients" for the double donut).
• They used their new method (building many different slides) to check if the fingerprint scanner worked.
• The Outcome: Their old method (using just a few slides) could prove the scanner worked for about 2,600 pairs of curves. Their new method (using hundreds of different slides) proved it for 3,282 pairs.
• They even found cases where the old method failed completely, but the new method succeeded.

Why This Matters

1. Progress on a 40-Year-Old Mystery: Beilinson's Conjecture is a huge open problem. Proving it for any new class of shapes is a big deal. This paper proves it for a much larger group of shapes than before.
2. A New Toolkit: They gave mathematicians a "recipe" to generate infinite families of these special curves. Before this, finding them was like finding a needle in a haystack; now, they have a machine to make the needles.
3. The "Rational" Hope: The paper suggests that while the world of complex numbers is chaotic and full of confusion, the world of rational numbers (fractions) is surprisingly orderly. The "slippery" points that confuse the scanner in the complex world simply don't exist when you stick to fractions.

Summary in One Sentence

The authors built a massive collection of special geometric "slides" to prove that for certain complex shapes, the rules of rational numbers are so strict that they force a perfect order, solving a major piece of a decades-old mathematical mystery.

Technical Summary

Here is a detailed technical summary of the paper "Hyperelliptic Curves Mapping to Abelian Varieties and Applications to Beilinson's Conjecture for Zero-Cycles" by Evangelia Gazaki and Jonathan Love.

1. Problem Statement

The paper addresses Beilinson's Conjecture regarding zero-cycles on smooth projective varieties defined over the rational numbers $\mathbb{Q}$.

• The Conjecture: For a smooth projective variety $X/\mathbb{Q}$, the Albanese map is injective. Equivalently, the kernel of the Albanese map, denoted $F^2(X)$, is zero.
• The Context: Over algebraically closed fields (like $\mathbb{C}$), $F^2(X)$ is known to be "enormous" and non-parametrizable if the geometric genus $p_g(X) > 0$. However, over $\mathbb{Q}$, it is conjectured that $F^2(X) = 0$.
• The Challenge: Proving this for surfaces with $p_g > 0$ is extremely difficult. No single example of a surface with $p_g > 0$ satisfying the conjecture was known prior to this work.
• Target Objects: The authors focus on abelian surfaces ($A$) and their associated Kummer surfaces ($K = \text{Kum}(A)$). For an abelian surface, $F^2(A)$ is generated by zero-cycles of the form $z_{a,b} = [a+b] - [a] - [b] + [0]$. The goal is to show these vanish for all $a, b \in A(\mathbb{Q})$.

2. Methodology

The authors develop a strategy linking the geometry of hyperelliptic curves mapping to abelian varieties with the vanishing of specific zero-cycles.

A. Hyperelliptic Points and Rational Equivalence

• Definition: A point $a \in A(k)$ is hyperelliptic if a nonzero multiple of it lies in the image of a morphism $f: H \to A$ from a hyperelliptic curve $H$, such that the negation map on $A$ induces the hyperelliptic involution $\iota$ on $H$ (i.e., $f \circ \iota = -f$).
• Key Lemma: If $a$ is a hyperelliptic point, then the zero-cycle $z_{a,a} = 0$ in $F^2(A)$.
• Bilinearity: The map $(a, b) \mapsto z_{a,b}$ is bilinear. Therefore, if a finite set of linear combinations of points are hyperelliptic, one can deduce that $z_{c,d} = 0$ for all $c, d$ in the divisible hull of the subgroup generated by those points.
• Reduction: To prove Beilinson's conjecture for $A/\mathbb{Q}$, it suffices to show that for any finite extension $L/\mathbb{Q}$, a specific finite set of points in $A(L)$ (derived from the Mordell-Weil basis) are hyperelliptic.

B. Construction via Kummer Surfaces and Elliptic Fibrations

The core geometric innovation is constructing a vast collection of such hyperelliptic curves.

1. Kummer Surface Structure: For an abelian surface $A$ isogenous to a product of elliptic curves $E \times E'$, the associated Kummer surface $K$ admits an elliptic fibration (Inose's pencil) $\xi: K \to \mathbb{P}^1$.
2. Mordell-Weil Lattice: The sections of this fibration form a Mordell-Weil lattice. By taking linear combinations of generators in this lattice, the authors produce infinitely many rational curves (sections) in $K$.
3. Pullback Construction: Pulling back a rational section $\phi: \mathbb{P}^1 \to K$ along the quotient map $E \times E' \to (E \times E')/\langle -1 \rangle$ yields a curve $C$. The normalization $H$ of $C$ is a hyperelliptic curve mapping compatibly to $E \times E'$.
4. Genus Control: By varying the coefficients in the Mordell-Weil lattice, the authors can control the genus of $H$. They prove that for infinitely many integers $g \ge 2$, there exist infinitely many non-isomorphic hyperelliptic curves of genus $g$ mapping to $A$.

C. Isogeny Invariance

A critical technical result (Proposition 3.21) proves that if a map $f: H \to A$ is "compatible" (satisfying the involution condition), then for any isogeny $\mu: A \to B$, the composition $\mu \circ f$ remains compatible. This ensures that the constructed curves work even when $A$ is not exactly a product $E \times E'$ but is isogenous to one.

3. Key Contributions and Results

Theoretical Results

• Theorem 1.2 / 2.6: A generalized criterion showing that if a finite set of linear combinations of points are hyperelliptic, then the entire subgroup generated by them has trivial Albanese kernel.
• Theorem 1.3 / 3.2: For an abelian surface $A$ isogenous to $E \times E'$, there exist infinitely many non-isomorphic hyperelliptic curves of infinitely many different genera mapping to $A$. Crucially, these curves are "independent" in the sense that no two have images related by multiplication maps.
• Higher Dimensions: The authors extend their results to abelian varieties of dimension $d \ge 3$ (Theorem 4.6), though they note the construction of sufficient hyperelliptic curves is more difficult in higher dimensions due to the lack of rational curves on general Kummer varieties.

Computational Results (Section 3.6)

The authors apply their theory to verify Beilinson's conjecture for specific pairs of elliptic curves over $\mathbb{Q}$.

• Previous Work: In a prior paper ([GL24]), they used genus 2 curves to prove finiteness of the relevant subgroup for 2,602 pairs of rank 1 elliptic curves.
• New Results: By utilizing:
  1. Higher genus curves (genus 6, 10, etc.) derived from higher sections of the Mordell-Weil lattice.
  2. Isogenous curves (replacing $E$ and $E'$ with isogenous curves to access more sections).
  • They increased the number of verified pairs significantly. For rank 1 pairs, they verified finiteness for 3,282 pairs (an increase of 680).
  • For rank 2 pairs, they verified finiteness for 1,221 pairs (an increase of 226).
• Significance of Computation: They found that for many pairs, genus 2 curves were insufficient to generate enough relations, but genus 6 or 10 curves provided the necessary independent relations to prove the zero-cycle vanishing.

4. Significance

• First Progress on Beilinson's Conjecture for Surfaces: This is the first substantial progress toward proving Beilinson's conjecture for surfaces with positive geometric genus ($p_g > 0$). Previously, the conjecture was open for all such surfaces.
• New Geometric Tool: The paper establishes a robust method for generating "infinite families" of hyperelliptic curves on abelian surfaces, moving beyond the limited genus 2 constructions previously known.
• Algorithmic Verification: The work provides an algorithmic framework to computationally verify the conjecture for specific number fields, demonstrating that the conjecture holds for a large (though finite) set of explicit examples.
• Connection to Bogomolov's Conjecture: The authors link their results to Bogomolov's conjecture (that points on K3 surfaces lie on rational curves). They show that if Bogomolov's conjecture holds for the Kummer surface, Beilinson's conjecture follows for the abelian surface. Their construction provides a concrete realization of this link over $\mathbb{Q}$.

Conclusion

Gazaki and Love successfully bridge the gap between the geometry of hyperelliptic curves and the arithmetic of zero-cycles. By constructing an abundant supply of hyperelliptic curves mapping to abelian surfaces, they prove that the Albanese kernel vanishes for a large class of points, providing the first concrete evidence supporting Beilinson's conjecture for surfaces with $p_g > 0$. Their work suggests that the conjecture is likely true for all abelian surfaces isogenous to products of elliptic curves.