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Algebraic cycles and values of Green's functions -- Products of Elliptic Curves

This paper establishes a connection between motivic cycles in the universal family of products of elliptic curves and Borcherds lifts of weakly holomorphic modular forms, thereby providing a motivic interpretation of the latter and proving Zagier's conjecture on higher Green's functions in cases where two CM points share the same discriminant.

Original authors: Ramesh Sreekantan (with an appendix by Kannappan Sampath)

Published 2026-06-23

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CC BY 4.0

Original authors: Ramesh Sreekantan (with an appendix by Kannappan Sampath)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Mathematical Treasure Hunt

Imagine you are a detective trying to solve a mystery about numbers. In the world of advanced math, there are special points called CM points (Complex Multiplication points). Think of these as "golden coordinates" on a map of shapes called elliptic curves.

For decades, mathematicians had a hunch (a conjecture by Gross and Zagier) that if you measure the "distance" or "energy" between two of these golden coordinates using a specific tool called a Green's function, the result would always be the logarithm of a "special" number (an algebraic number). It's like predicting that if you measure the distance between two specific cities, the answer will always be a number you can write down using simple fractions and roots, never a messy, infinite decimal.

This paper confirms that hunch for a specific, tricky scenario. But instead of just measuring the distance, the author builds a bridge between two different worlds of math to prove it.

The Two Worlds: The "Map" and the "Machine"

The paper connects two very different ways of looking at the same problem:

The Automorphic World (The Machine): This approach uses "weakly holomorphic modular forms." Imagine these as complex, vibrating musical instruments. When you pluck a string (perform a calculation), it produces a sound (a Green's function). Previous researchers proved the treasure hunt result by analyzing the sound waves directly.
The Motivic World (The Map): This approach uses "motivic cycles." Imagine these as physical roads or bridges built between the golden coordinates. The author constructs these roads out of algebraic shapes.

The Author's Breakthrough:
Sreekantan says, "Let's build a physical road (a motivic cycle) between these points and see what happens when we drive over it." He constructs these roads on a special surface called a Kummer surface (which is like a folded, crumpled sheet of paper that has been smoothed out).

The Key Analogy: The "Indecomposable" Bridge

To prove the result, the author needs to build a very specific type of bridge.

Decomposable bridges are like bridges made of two separate, unconnected planks. They are boring and don't tell us much new.
Indecomposable bridges are like a single, solid, twisted arch that cannot be taken apart. These are rare and hard to build.

The author successfully builds these "indecomposable" bridges in a family of shapes. He proves that these bridges are "real" and not just illusions.

The "Boundary" Test

How do we know the bridge is real? The author uses a "boundary test."
Imagine your bridge spans a river. If you look at the ends of the bridge where it touches the shore, you see specific patterns (Hecke cycles).

The author shows that the "ends" of his newly built bridge match the patterns predicted by the theory.
Because the ends match the theory, the bridge itself must be a valid, indecomposable structure.

The Payoff: Why the Distance is a "Special" Number

Once the bridge is built, the author drives a car over it (calculates a "regulator").

The Result: The car's odometer reads a number.
The Magic: The author proves that this odometer reading is exactly the same as the "Green's function" value (the distance/energy) we were trying to measure.
The Conclusion: Because the bridge is made of algebraic materials (rational functions on algebraic curves), the odometer reading must be the logarithm of an algebraic number.

In simple terms: Because we built the bridge out of "nice" algebraic blocks, the measurement taken while crossing it must also be a "nice" number.

The "Dictionary" Between Worlds

The paper suggests a fascinating dictionary between the two worlds mentioned earlier:

Every "musical instrument" (weakly holomorphic modular form) has a corresponding "bridge" (motivic cycle).
The sound the instrument makes is the same as the measurement taken on the bridge.

This implies that the mysterious, vibrating sounds of modular forms actually have a physical, geometric shape behind them.

Summary of What Was Proven

The Conjecture: The author proves that for two specific types of "golden coordinates" (CM points) that share the same underlying properties, the value of the Green's function is indeed the logarithm of an algebraic number.
The Method: He didn't just calculate the value; he constructed a geometric object (a motivic cycle) that forces the value to be algebraic.
The Connection: He showed that the geometric "bridges" he built correspond perfectly to the "musical instruments" (modular forms) used by other mathematicians to solve similar problems.

What the paper does NOT do:

It does not apply this to physics, engineering, or medicine.
It does not claim to solve the Riemann Hypothesis.
It does not claim this works for every possible pair of points, only for specific cases where the points share the same "discriminant" (a specific mathematical fingerprint).

In essence, the paper is a masterclass in building a geometric bridge to prove that a complex mathematical measurement is actually made of simple, clean ingredients.

Technical Summary: Algebraic Cycles and Values of Green's Functions: Products of Elliptic Curves

Problem Statement
The paper addresses the Gross-Zagier conjecture regarding the values of "higher Green's functions" at Complex Multiplication (CM) points. Specifically, Gross and Zagier conjectured that for a weakly holomorphic modular form $f$ of weight $-2j$ , the value of the associated higher Green's function $G^f_{j+1}(z_1, z_2)$ at a pair of CM points $(z_1, z_2)$ is the logarithm of an algebraic number (up to a specific discriminant factor). While this conjecture has recently been proven as a theorem by Li [Li23] and Bruinier-Li-Yang [BLY25] using Borcherds lifts and automorphic methods, this paper seeks to establish the result through a distinct, motivic framework. The authors aim to relate the algebraicity of these values to the existence of specific "motivic cycles" in the universal family of products of elliptic curves.

Methodology
The paper employs a two-pronged approach combining algebraic geometry (specifically the theory of motivic cohomology and regulators) with the arithmetic of modular curves.

Construction of Motivic Cycles:
- The authors work within the universal family of products of elliptic curves with full level 2 structure, denoted $A \to X(2) \times X(2)$ .
- They focus on the associated Kummer K3 surface, $\tilde{K}_{A_\eta}$ , obtained by blowing up the nodes of the Kummer surface of the generic fiber.
- Using enumerative geometry of rational curves in $\mathbb{P}^1 \times \mathbb{P}^1$ , they construct explicit elements in the motivic cohomology group $H^3_M(\tilde{K}_{A_\eta}, \mathbb{Q}(2))$ .
- The construction relies on deforming the graph of an isogeny $\phi: E_1 \to E_2$ of odd degree $n$ . This graph, when projected to the Kummer surface and further to $\mathbb{P}^1 \times \mathbb{P}^1$ , yields a rational curve of bidegree $(n, 1)$ .
- By analyzing the intersection of these curves with the ramification lines of the Kummer map, the authors define a cycle $\xi$ supported on the generic fiber. They prove this cycle is indecomposable (i.e., not in the image of the product map from lower weight cohomology) by computing its "non-Archimedean" regulator (the boundary map in the localization sequence). The boundary is shown to be supported on Hecke correspondences $T_n$ , specifically taking the form of the difference between the closures of the graphs of $\phi$ and $-\phi$ .
Regulator Computation and Algebraicity:
- The authors compute the Archimedean regulator of these constructed cycles against a specific $(j, j)$ -form $\eta^j_{z_1, z_2}$ on the fiber over a pair of points $(z_1, z_2)$ .
- They establish a link between this regulator and the higher Green's functions $G_{j+1}$ . Specifically, the regulator pairing is shown to be proportional to a linear combination of Green's functions evaluated at the points.
- When $(z_1, z_2)$ are CM points with the same discriminant, the form $\eta^j_{z_1, z_2}$ is represented by an algebraic cycle (a CM cycle).
- By the compatibility of the regulator with the intersection pairing in motivic cohomology, the value of the regulator at these CM points reduces to the logarithm of the absolute value of a rational function evaluated at the intersection points of the motivic cycle and the CM cycle. Since all data is defined over $\overline{\mathbb{Q}}$ , the result is the logarithm of an algebraic number.

Key Contributions and Results

Explicit Construction of Indecomposable Cycles: The paper provides an explicit construction of indecomposable motivic cycles in $H^3_M(\tilde{K}_{A_\eta}, \mathbb{Q}(2))$ for the universal family of products of elliptic curves. These cycles are constructed via the enumerative geometry of rational curves in $\mathbb{P}^1 \times \mathbb{P}^1$ and are shown to have boundaries supported on Hecke correspondences.
Proof of Algebraicity for Same-Discriminant CM Points: The authors prove that for two CM points $z_1, z_2$ $z_{1}, z_{2}$ with the same discriminant $D$ $D$ , the value of the degree 2 higher Green's function $G_2(z_1, T_n z_2)$ $G_{2} (z_{1}, T_{n} z_{2})$ (where $T_n$ $T_{n}$ is a Hecke correspondence) is the logarithm of an algebraic number.
- Theorem 1.4 / Theorem 6.2: $2Dn G_2(z_1, T_n z_2) = \log |\alpha|$ for some $\alpha \in \overline{\mathbb{Q}}$ .
- This recovers a result originally proven by Viazovska [Via15] for the case of same discriminants, but via a motivic approach rather than Borcherds lifts.
Generalization to Mixed Discriminants (Conditional): The paper notes that while the motivic approach naturally handles same-discriminant cases (where the CM cycle exists in the fiber), it suggests a path toward mixed discriminant cases via Arakelov pairings, though this is not fully resolved in the current work.
Link Between Motivic Cycles and Borcherds Lifts: The paper proposes a conjectural dictionary (Conjecture 1.5 and Conjecture 7.1) linking weakly holomorphic modular forms to motivic cycles. It suggests that the regulator of a motivic cycle $\xi_f$ associated with a modular form $f$ coincides with the Borcherds lift $\Phi(f)$ of that form. This provides a "motivic interpretation" of weakly holomorphic modular forms.

Significance
The paper claims significance primarily in establishing a bridge between two previously distinct approaches to the Gross-Zagier conjecture:

The Motivic Approach: Following Zhang [Zha97] and Mellit [Mel08], which links Green's functions to regulators of algebraic cycles.
The Automorphic Approach: Following Li, Bruinier, and Yang, which links Green's functions to Borcherds lifts.

By constructing explicit cycles and computing their regulators, the authors demonstrate that the algebraicity of Green's function values is a direct consequence of the existence of these motivic cycles. The work suggests that the "modular complex" (a subcomplex of motivic cohomology generated by special cycles) contains the necessary elements to explain the algebraicity of automorphic Green's functions. The paper positions itself as a step toward a unified understanding where the "motivic Green's function" (regulator of a cycle) and the "automorphic Green's function" (Borcherds lift) are identified.

The authors remain modest regarding the scope, noting that their explicit results currently cover the case of products of elliptic curves and CM points with the same discriminant. They acknowledge that the general case (different discriminants) and higher dimensions (e.g., $G_3$ ) require further development, which they outline as directions for future work (e.g., in companion papers [Sre22], [Sre26]).