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Gross's conjecture: the dihedral case

This paper proves Gross's 2000 conjecture that the Fourier coefficients of quaternionic modular forms on $\mathsf{G}_2$ encode $L$ -values of cubic twists for the specific case where the associated modular forms are dihedral, thereby providing the first known examples of this phenomenon.

Original authors: Petar Bakić, Aleksander Horawa, Siyan Daniel Li-Huerta, Naomi Sweeting

Published 2026-07-07

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CC BY 4.0

Original authors: Petar Bakić, Aleksander Horawa, Siyan Daniel Li-Huerta, Naomi Sweeting

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 the world of mathematics as a vast, intricate library. Inside this library, there are special books called modular forms. These aren't books you read for a story; they are complex mathematical objects that act like "universal translators." They hold secret codes that connect different areas of math, specifically linking shapes, numbers, and the behavior of prime numbers.

This paper is about a specific, very rare, and difficult-to-find type of book in this library: Quaternionic Modular Forms on $G_2$ .

Here is the story of what the authors did, explained simply:

1. The Mystery of the Hidden Codes

For a long time, mathematicians knew that some of these "books" (modular forms) had a special feature: if you opened them up, you could find a list of numbers inside called Fourier coefficients.

A famous mathematician named Waldspurger discovered a magic trick with a different type of book. He found that the numbers in the list were actually hiding the answers to specific questions about other numbers (called L-values). It was like finding a treasure map where the "X" marked the spot of a hidden value.

In 2000, another giant of the field, Gross, made a bold guess (a conjecture). He said: "If you look at these specific, rare books called Quaternionic Modular Forms on $G_2$ , their hidden number lists should also be connected to a different kind of treasure map. Specifically, they should reveal the values of 'cubic twists' of other modular forms."

For 25 years, no one could prove this. It was like knowing the treasure map existed but being unable to read the handwriting.

2. The "Dihedral" Key

The authors of this paper decided to tackle this mystery, but they knew the whole library was too big to search at once. So, they decided to look at just one specific section of the library first. They focused on a special category of these books called the "Dihedral Case."

Think of "Dihedral" as a specific flavor or style of these mathematical objects. It's a simpler, more structured version of the general problem, like solving a puzzle with a picture on the box before trying to solve the puzzle blindfolded.

The Goal: Prove that Gross was right, but only for this specific "Dihedral" flavor.

3. The Method: Building a Bridge

To prove the connection, the authors had to build a bridge between two distant islands:

Island A: The Quaternionic Modular Forms on $G_2$ (the rare books with the hidden codes).
Island B: The "Cubic Twists" (the treasure values Gross predicted).

They used a mathematical tool called a Theta Lift. Imagine this as a high-tech conveyor belt or a translation machine.

They started with a known object (a modular form on a group called $PU_3$ ).
They ran it through their "Theta Lift" machine.
The machine spat out a Quaternionic Modular Form on $G_2$ .

Now, they had a specific book to examine.

4. The Detective Work: Reading the Codes

Once they had their specific book, they needed to read the hidden numbers (the Fourier coefficients). This is where the math gets very technical, but the idea is simple:

The Local vs. Global: The authors had to check the book's codes in two different ways.
- The "Local" Check: They looked at the codes in specific, small neighborhoods (like checking a single page under a microscope). They had to prove that for certain "neighborhoods" (called local places), the codes were either perfectly clear (value = 1) or completely blank (value = 0).
- The "Global" Check: They looked at the whole book at once. They calculated the total value of the codes.
The "Obstacle" Course: They found that the codes didn't always appear. Sometimes, the "local" conditions (the neighborhood rules) blocked the code from showing up. They had to prove exactly when the code would appear and when it would vanish. It was like realizing a treasure map only works if you are standing on the right type of soil.

5. The Big Reveal

After doing all this heavy lifting—calculating complex integrals, checking thousands of tiny cases, and building their bridge—they reached the conclusion:

Gross was right.

For the "Dihedral" cases, the hidden numbers in the Quaternionic Modular Forms do exactly what Gross predicted. They are directly proportional to the "L-values" (the treasure values) of the cubic twists.

The Takeaway

Think of this paper as the first time someone successfully cracked a safe that had been locked for 25 years.

Before: We knew the safe existed and had a theory about what was inside (Gross's Conjecture).
During: The authors built a specialized tool (the Theta Lift) and a set of keys (the local integrals) to open it.
After: They opened the safe, found the treasure (the L-values), and proved the theory works for this specific type of safe.

They didn't open every safe in the library yet (the general case is still unsolved), but they proved the theory works for the first, most important class of examples. This gives mathematicians confidence that the theory is true for the rest of the library, too.

In short: They proved a 25-year-old guess about how two complex mathematical worlds talk to each other, but only for a specific, well-behaved family of numbers. They showed that the "secret codes" in these rare mathematical objects are indeed the keys to unlocking deep arithmetic secrets.

Technical Summary: Gross's Conjecture: The Dihedral Case

Problem Statement
The paper addresses a conjecture proposed by Gross in 2000 regarding the arithmetic structure of quaternionic modular forms on the split simple group $G_2$ . While Waldspurger established a profound relationship between the Fourier coefficients of half-integral weight modular forms and the $L$ -values of quadratic twists of integral weight forms, Gross conjectured an analogous relationship for quaternionic modular forms on $G_2$ . Specifically, he predicted that the Fourier coefficients of these forms, which are indexed by totally real cubic rings, encode the central $L$ -values ( $s=1/2$ ) of cubic twists of associated holomorphic modular forms on $PGL_2$ .

Prior to this work, the conjecture was known only in specific, non-cuspidal cases (e.g., Eisenstein series) or for very restricted cubic algebras. The primary challenge lay in the lack of cuspidal examples where the conjecture could be verified, particularly because the associated modular forms often have non-squarefree levels, complicating the application of existing theta lift techniques.

Methodology
The authors prove the conjecture for the "dihedral case," where the underlying holomorphic modular form $f$ is associated with a conjugate-symplectic Hecke character $\chi$ of an imaginary quadratic field $K$ . The methodology relies on the construction of quaternionic modular forms via an exceptional theta lift and the subsequent explicit computation of their Fourier coefficients.

Theta Lift Construction: The authors construct the cuspidal quaternionic modular form $F_\epsilon$ on $G_2$ as a theta lift from a compact unitary group $G' = PU_3$ (associated with $K$ ) to $G_2$ . This lift utilizes theta kernels on a quasi-split adjoint form of $E_6$ . The representation $\pi_\epsilon$ of $G_2$ is associated with a cuspidal automorphic representation $\sigma_\epsilon$ of $G'$ via Arthur's conjectures, which the authors had previously established unconditionally for this case.
Fourier Coefficient Analysis: The Fourier coefficient $a_E(F_\epsilon)$ is related to an automorphic period on $G'$ . By analyzing the theta kernel along the unipotent radical of the Heisenberg parabolic of $E_6$ , the authors derive an integral formula (Equation 1.2) expressing the Fourier coefficient as a product of a global torus period $P_i(f')$ on $G'$ and a product of local integrals $I_v$ .
Seesaw Identity and Periods: A "seesaw" diagram involving unitary groups over $K$ and cubic algebras $E$ is employed to relate the square of the global period $|P_i(f')|^2$ to the $L$ -value $L(1/2, f \otimes V_E)$ , where $V_E$ is the 2-dimensional Artin representation associated with the cubic algebra $E$ . This step utilizes explicit calculations by T. Yang regarding theta lifts from unitary groups to orthogonal groups.
Local Integral Computation: The core technical contribution involves the explicit computation of the local integrals $I_v(E_v, \phi_v, f'_v)$ $I_{v} (E_{v}, ϕ_{v}, f_{v}^{'})$ at all places:
- Unramified places ( $p \nmid N$ ): The authors utilize spherical vectors and prove that the integral is non-zero (and equals 1) only if the cubic algebra is unramified and maximal, reducing the problem to a group-theoretic statement about conjugates of algebra embeddings (Lemma 4.5).
- Ramified places ( $p \mid N$ ): The authors construct specific "test vectors" (local vectors $\phi_p$ and $f'_p$ ) such that the local integral acts as a characteristic function: it equals 1 if the local sign condition $\epsilon_p = \epsilon_p(E_p, \chi_p)$ is satisfied, and 0 otherwise. This resolves the local obstructions that prevent the Fourier coefficient from vanishing.
- Archimedean places: The computation relies on the theory of quaternionic discrete series and the minimal representation of $E_6$ . The authors define specific raising operators to construct highest weight vectors and compute the resulting integrals using Bessel functions and properties of the Freudenthal construction.

Key Contributions and Results
The paper establishes the first unconditional proof of Gross's conjecture for a class of cuspidal quaternionic modular forms.

Main Theorem (Theorem 1.2 / Theorem 3.14): For a dihedral modular form $f$ of even weight $2k$ associated with a Hecke character $\chi$ (with $L(1/2, \chi) \neq 0$ ), there exists a quaternionic modular form $F_\epsilon$ on $G_2$ such that for any totally real étale cubic algebra $E/\mathbb{Q}$ with ring of integers $\mathcal{O}_E$ :
$|a_E(F_\epsilon)|^2 = L(1/2, f \otimes V_E) \cdot \Delta_E^{(k-1)/2}$
provided that local sign conditions $\epsilon_p = \epsilon_p(E_p, \chi_p)$ are met for all primes $p$ dividing the level of $f$ . If these conditions are not met, the Fourier coefficient vanishes.
Explicit Local Integrals: The paper provides a complete evaluation of the local integrals arising from the theta lift, including a novel construction of test vectors at ramified primes that isolate specific local signs.
Generalization to Totally Real Fields: While the introduction focuses on $\mathbb{Q}$ , the authors note that their methods generalize to totally real fields $F$ and dihedral Hilbert modular forms (Theorem 3.14).

Significance
The paper claims significance as the first examples for which Gross's conjecture is known to hold. It bridges the gap between the theory of quaternionic modular forms on $G_2$ and the arithmetic of cubic twists of modular forms.

The authors emphasize that their result is consistent with Pollack's earlier work showing that Fourier coefficients of such forms lie in abelian extensions ( $\mathbb{Q}^{ab}$ ). By relating the coefficients to $L$ -values of twists of dihedral forms (which are known to have periods in $\mathbb{Q}^{ab}$ via Blasius's work), the paper confirms the arithmetic nature of these coefficients.

The work also clarifies the role of local obstructions: unlike the classical Shimura lift context where coefficients are non-zero for all quadratic twists, the quaternionic case requires specific local sign conditions ( $\epsilon_p$ ) to be satisfied for the Fourier coefficients to be non-vanishing. This paper provides the precise mechanism (via the choice of test vectors in the theta lift) to satisfy these conditions and extract the $L$ -values.

The authors maintain a modest tone regarding the scope, noting that their results apply specifically to the dihedral case and that the general case for non-dihedral forms remains open. They do not claim to resolve the full conjecture for all modular forms but rather to provide the foundational proof for the first non-trivial class of examples.