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A higher Gross-Zagier type formula for moduli of shtukas at deeper level

This paper generalizes the Yun-Zhang function field analogue of the Gross-Zagier and Waldspurger formulae to moduli spaces of shtukas with non-square-free (deeper) levels by analyzing the geometry and cohomology of their integral models.

Original authors: Patrick Bieker

Published 2026-07-09

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Patrick Bieker

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

Imagine you are trying to solve a massive, cosmic puzzle where two very different worlds are secretly connected. On one side, you have numbers and equations (specifically, things called "L-functions" that describe the hidden rhythms of prime numbers). On the other side, you have shapes and geometry (specifically, complex, multi-dimensional landscapes called "moduli spaces" that act like maps for these numbers).

For a long time, mathematicians had a special rule, a "Rosetta Stone," that translated between these two worlds. This rule is called the Gross-Zagier formula. It says: If you look at how a specific number-pattern changes (its derivative), it tells you exactly how much "space" a specific geometric shape occupies on its map.

However, this rule only worked when the puzzle pieces were "square-free." Think of this like a puzzle where every piece is a unique, single block. If you tried to use pieces that were glued together in clusters (what mathematicians call "deeper level" or "non-square-free"), the old map didn't work. The geometry got messy, and the translation broke down.

Patrick Bieker's paper is about fixing the map for the messy, clustered pieces.

Here is how he did it, using some everyday analogies:

1. The Problem: The "Deep" Level

In the world of these mathematical puzzles, "level" refers to how tightly the pieces are packed.

Square-free level: Like a grid where every spot holds exactly one tile. The rules are clean and simple.
Deeper level: Like a grid where some spots hold stacks of tiles, or tiles are glued together in complex ways.

Previous mathematicians (Yun and Zhang) had built a perfect bridge between the number-world and the shape-world for the simple, single-tile grids. But when they tried to cross over to the "stacked tile" (deeper level) areas, the bridge collapsed. They didn't have a good "integral model"—a sturdy, continuous map that could handle the complexity of these stacked tiles without falling apart.

2. The Solution: A New, "Naive" Map

Bieker's main achievement is building a new kind of map for these deeper levels.

The Old Approach: Trying to use a very complex, high-tech construction (Drinfeld level structures) that was too heavy and complicated for these specific deep layers. It was like trying to use a supercomputer to count apples in a basket; it worked for some things, but it got confused when the apples were piled up.
Bieker's Approach: He used a "naive" (simple) approach. He realized that for the specific geometric shapes he needed to measure (called Heegner-Drinfeld cycles), he didn't need the whole complex map. He only needed the "smooth, open" parts of the map.

Think of it like this: If you want to measure the area of a garden, you don't need to map every single weed in the soil. You just need the clear, open grass. Bieker defined a new, simpler moduli space (a "naive" map) that captures exactly the parts of the geometry where the special shapes live, ignoring the messy, crowded parts that don't matter for the calculation.

3. The Result: The Formula Works Again

Once he built this new, sturdy map for the deeper levels, he could run the translation again.

He proved that the Gross-Zagier formula still holds true, even when the puzzle pieces are stacked and complex.
He showed that the "derivative" of the number-pattern (how fast the rhythm changes) is still perfectly equal to the "intersection number" (how the geometric shapes overlap) on this new map.

4. Why It Matters (According to the Paper)

The paper suggests that this breakthrough is a stepping stone. By proving that this deep-level formula works, it opens the door to understanding the Birch and Swinnerton-Dyer conjecture in the "function field" setting (a specific type of number system) for cases that are more complex than just the simplest scenarios (rank 1).

In summary:
Bieker took a famous mathematical rule that only worked on simple, clean surfaces and successfully extended it to work on complex, layered surfaces. He did this by inventing a new, simpler way to draw the map of the complex surface, proving that the deep, hidden connection between numbers and shapes remains unbroken, even in the most tangled parts of the mathematical universe.

Technical Summary: A Higher Gross-Zagier Type Formula for Moduli of Shtukas at Deeper Level

Problem Statement
The paper addresses the extension of the higher-order Gross-Zagier and Waldspurger formulae, originally established by Yun and Zhang for function fields with square-free level structures, to the case of non-square-free (deeper) levels.

In the seminal works [YZ17, YZ19], Yun and Zhang related the higher-order central derivatives of base change $L$ -functions of cuspidal automorphic representations to the self-intersection numbers of Heegner-Drinfeld cycles on moduli spaces of $G$ -shtukas (where $G = \text{PGL}_2$ ). These results were restricted to square-free level structures $\Sigma$ . The authors note that extending these results to deeper levels was previously considered "out of reach" due to the lack of an appropriate "integral model" of the moduli space of shtukas that could support the necessary special cycles (Heegner-Drinfeld cycles) in the presence of multiple colliding legs or higher multiplicities in the level divisor.

Methodology and New Input
The primary methodological innovation is the introduction and rigorous study of moduli spaces of shtukas with "naive" $\Gamma_0(\Sigma)$ -level structures for non-reduced divisors $\Sigma$ .

Naive Integral Models: Instead of attempting to construct a full Drinfeld level structure analogue (which becomes problematic when legs collide or multiplicities exceed one), the author defines the moduli space $\text{Sht}^r_G(\Sigma; \Sigma_\infty)$ as a moduli space of shtukas for a smooth affine group scheme encoding the $\Gamma_0(\Sigma)$ -level structure.
- The author acknowledges that for non-square-free levels, this "naive" model corresponds to an open substack of the more intricate integral models defined via Drinfeld level structures (specifically, the smooth locus of the special fiber).
- Crucially, the paper argues that the Heegner-Drinfeld cycles are supported entirely on this open substack, rendering the missing components (such as supersingular points and other components appearing in the Drinfeld model) irrelevant for the intersection number calculations.
Geometric Analysis: The paper establishes the geometric properties of these new moduli spaces:
- Smoothness: The moduli space $\text{Sht}^r_G(\Sigma; \Sigma_\infty)$ is proven to be smooth over the base field $k$ of dimension $2r$ .
- Flatness: The projection to the legs $X^r \times S_\infty$ is flat of relative dimension $r$ and smooth over the locus where the level is Iwahori (reduced).
- Hecke Correspondences: The author extends the definition of Hecke correspondences and the action of the spherical Hecke algebra to these deeper level spaces, proving they act on the Chow groups and cohomology.
Automorphic and Spectral Tools:
- Modified Eisenstein Ideal: A new "modified Eisenstein ideal" $\tilde{I}^\Sigma_{\text{Eis}}$ is defined within the Hecke algebra to account for the multiplicities in the divisor $\Sigma$ . This ideal replaces the standard Eisenstein ideal used in the square-free case.
- Local Spherical Characters: The author computes local spherical characters at places with deeper level structures (where the conductor exponent $n_x \geq 2$ ), utilizing specific test functions (characteristic functions of $\Gamma_0(\varpi^{n_x}_x)$ ) and Whittaker models.
- Relative Trace Formula: The paper employs Jacquet's relative trace formula, relating the global spherical character (and its derivatives) to orbital integrals.
- Geometrization: Orbital integrals are geometrized as traces of Frobenius on a Hitchin-type fibration, generalizing the construction in [YZ19] to the deeper level setting.

Key Results

Theorem 1.1 (Main Formula): The paper establishes a higher Gross-Zagier type formula for non-square-free levels. Let $\pi$ be a cuspidal automorphic representation of $\text{PGL}_2(\mathbb{A}_F)$ with conductor $\Sigma$ , and let $\nu: X' \to X$ be a quadratic cover satisfying the generalized Heegner hypothesis. Then:
$|\omega_X| q^{\rho/2 - N} \cdot r! \cdot (-\log q)^r \cdot L(\pi, \text{Ad}, 1) \cdot L^{(r)}_{F'/F}(\pi, 0) = (Z^\mu_\pi(\xi), Z^\mu_\pi(\xi))_{\text{Sht}^r_G(\Sigma; \Sigma_\infty)}$
Here, the left side involves the $r$ -th derivative of the base change $L$ -function at the central point, normalized by local factors and the degree of the canonical bundle. The right side is the self-intersection number of the $\pi$ -isotypic component of the Heegner-Drinfeld cycle on the moduli space of shtukas with deeper level.
Theorem 1.2 (Geometry): Confirms that the moduli space $\text{Sht}^r_G(\Sigma; \Sigma_\infty)$ is a smooth Deligne-Mumford stack of dimension $2r$ , and the projection to the legs is flat and smooth over the reduced locus.
Theorem 1.3 (Spectral Decomposition): Proves a spectral decomposition of the middle-degree compactly supported cohomology $H^{2r}_c$ under the action of the Hecke algebra. The cohomology decomposes into a finite-dimensional cuspidal part and a part supported on the modified Eisenstein ideal, generalizing the results of [YZ19] to deeper levels.

Significance and Claims
The paper claims to successfully generalize the Yun-Zhang formulae to not necessarily square-free levels, thereby removing a significant restriction in the function field analogue of the Gross-Zagier formula.

Resolution of the "Integral Model" Obstacle: The author claims that the lack of a suitable integral model was the primary barrier to this generalization. By introducing the "naive" moduli space and demonstrating that it suffices for the intersection theory of Heegner-Drinfeld cycles, the paper overcomes this obstacle.
Consistency with Existing Theory: The results are shown to be consistent with the square-free case (where the naive model coincides with the standard one) and the classical Gross-Zagier formula for modular curves (where the cycles are supported on the smooth locus).
Potential for BSD: The introduction notes that these higher Gross-Zagier formulas offer a pathway to gain new insight into the Birch and Swinnerton-Dyer (BSD) conjecture in the function field setting, potentially extending beyond the rank 1 case, though the paper focuses on the derivation of the formula itself rather than immediate arithmetic applications.

The work relies heavily on the strategy of Yun and Zhang, adapting their geometric and automorphic arguments to the specific technical challenges posed by deeper level structures, particularly the geometry of the moduli stacks and the computation of local factors.

1. The Problem: The "Deep" Level

2. The Solution: A New, "Naive" Map

3. The Result: The Formula Works Again

4. Why It Matters (According to the Paper)

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