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Beyond the Adams Conjecture

This paper demonstrates that for symplectic and even orthogonal groups over a $p$ -adic field, the local theta lifts of tempered representations at the first occurrence can belong to significantly more local Arthur packets than predicted by the Adams conjecture, by explicitly determining the number of such packets.

Original authors: Alexander Hazeltine, Aarya Kumar, Andrew Tung

Published 2026-07-28

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CC BY 4.0

Original authors: Alexander Hazeltine, Aarya Kumar, Andrew Tung

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 a vast, invisible universe made entirely of mathematical shapes and symmetries. In this world, mathematicians study "groups," which are like rulebooks for how objects can move, rotate, or transform without breaking. Think of a Rubik's Cube: you can twist its faces in specific ways, and the cube stays a cube. That's a group. Now, imagine trying to predict exactly what happens when you take a complex, twisting shape from one rulebook and translate it into a different, slightly larger rulebook. This is the job of the "Langlands program," a massive, decades-long quest to connect different areas of math like a universal translator.

One of the most famous tools in this quest is the "theta lift." Picture it as a magical elevator that takes a specific mathematical object (a representation) from a smaller building (a symplectic group) and moves it up to a taller building (an orthogonal group). For a long time, mathematicians believed this elevator was very predictable. They thought that if you put a specific object in, it would always land in a very specific, pre-assigned "neighborhood" (called an Arthur packet) in the new building. This idea was known as the "Adams conjecture." It was like believing that every time you mail a letter to a specific address, it always lands in the exact same mailbox, never missing or landing in a neighbor's.

However, recent discoveries suggested that the universe of math might be messier than that. Maybe the elevator doesn't just land in one mailbox; maybe it drops the letter into several different ones, some of which were never on the original map. This paper, titled "Beyond the Adams Conjecture," dives deep into this mystery. The authors, Alexander Hazeltine, Aarya Kumar, and Andrew Tung, set out to count exactly how many different "mailboxes" (local Arthur packets) a mathematical object can land in when it takes this elevator ride for the very first time it can go up. They aren't just guessing; they are building a precise, step-by-step counting machine to figure out the exact number of possibilities.

The paper's main finding is a new set of rules that tell us exactly how many different Arthur packets contain the "first occurrence" of a theta lift. The authors discovered that the old map (the Adams conjecture) was too simple. While the conjecture predicted that a lifted object would land in exactly one specific neighborhood, the authors prove that it often lands in many more. In fact, for certain types of mathematical objects, the number of possible landing spots can be three times larger than the old theory predicted. They didn't just find this out by accident; they developed a complex, recursive formula—a kind of mathematical recipe—that allows you to calculate this number based on the structure of the original object.

The authors are very careful to distinguish between what they have proven and what they are still guessing. They have rigorously proved their counting formula for six specific scenarios (cases) involving the structure of these mathematical objects. In these six cases, they can say with 100% certainty exactly how many packets the lift belongs to. However, there is one tricky scenario (Case 7) where the signs of the mathematical "circles" in the object are opposite to each other. For this specific case, the authors offer a conjecture—a very well-reasoned guess based on patterns they see—but they have not yet proven it. They provide strong motivation for why this guess should be true, but it remains a hypothesis until further work is done.

The paper also explicitly rules out the idea that the Adams conjecture holds true in all situations. They show that while the conjecture works perfectly when the "elevator" goes up a very long way (when the difference in size between the buildings is huge), it fails when the elevator only goes up a short distance. In these short trips, the object doesn't just land in the predicted packet; it spreads out into a whole family of packets. The authors use a clever system of "blocks" and "almost-blocks"—like stacking LEGO bricks in specific patterns—to break down these complex objects into manageable pieces. By analyzing how these blocks interact, they can count the possibilities.

In the end, this paper doesn't just correct a single prediction; it reveals a richer, more complex landscape for these mathematical objects. It shows that the "elevator" of the theta correspondence is far more adventurous than previously thought, dropping its passengers into a variety of neighborhoods that were previously overlooked. While the authors have solved the puzzle for most of the common scenarios, they leave one final piece of the puzzle (Case 7) as a challenge for future mathematicians, inviting others to finish the proof of their elegant, albeit incomplete, map.

Technical Summary: Beyond the Adams Conjecture

Problem Statement
This paper investigates the behavior of local theta lifts for symplectic and even orthogonal groups over a $p$ -adic field, specifically focusing on the "going-up" tower at the first occurrence. The central problem is to determine the number of local Arthur packets containing the theta lift of a tempered representation.

While the influential Adams conjecture predicts that the local theta correspondence preserves local Arthur packets (mapping a representation in packet $\Pi_\psi$ to one in $\Pi_{\psi_\alpha}$ ), it is known that this conjecture fails in general. Specifically, theta lifts often lie in local Arthur packets that are not predicted by the Adams conjecture (termed "non- $\theta$ -relevant" pairs). Previous work established that for sufficiently large rank differences ( $\alpha \gg 0$ ), the Adams conjecture holds, and for supercuspidal representations, the number of containing packets is known to be three times that of the original representation. However, a systematic understanding of the count $|\Psi(\theta_{-\alpha}^{\text{up}}(\pi))|$ for general tempered representations $\pi$ at the first occurrence remained incomplete.

Methodology
The authors employ the combinatorial framework of extended multi-segments to parameterize local Arthur packets, building upon the work of Atobe, Mœglin, and others. The methodology involves:

Reduction to Combinatorics: The problem of counting Arthur packets is reduced to a combinatorial counting problem involving the structure of extended multi-segments. The authors utilize the theory of intersections of local Arthur packets and the classification of packets via extended multi-segments (specifically those of "good parity").
Block Decomposition: Tempered extended multi-segments are decomposed uniquely into disjoint "blocks" and "almost-blocks." The paper relies on recursive formulas derived in prior work (specifically [20]) to compute the number of packets for individual blocks.
Operator Analysis: The authors analyze the action of specific operators (row exchange $R_k$ , union-intersection $u_{i,j}$ , dual operators, and raising operators) on these multi-segments. They study how the theta lift operation $\Theta_1$ (representing the lift to the first occurrence) interacts with these operators and the block structure.
Case-by-Case Analysis: The main result is derived by classifying the interaction between the first two blocks of a tempered representation's decomposition. The analysis distinguishes seven cases based on the column indices, multiplicities, and sign patterns of the extended segments at the boundary of the first block.

Key Contributions and Results

Main Theorem (Theorem 4.11): The paper provides a recursive formula for computing $|\Psi(\theta_{-\alpha}^{\text{up}}(\pi))|$ $∣Ψ (θ_{- α}^{up} (π)) ∣$ for tempered representations $\pi$ $π$ in six out of seven possible cases. These cases are determined by the relationship between the last column of the first block ( $H_{\text{col}}$ $H_{col}$ ) and the first column of the second block ( $N_{\text{col}}$ $N_{col}$ ), as well as the multiplicities and signs of the segments in these columns.
- In cases where the theta lift remains tempered (Cases 1, 2, 4, 6), the count is derived directly from the block decomposition using existing formulas.
- In cases where the lift interacts with the second block (Cases 3 and 5), the authors derive new recursive relations. For instance, in Case 3, the count depends on the multiplicity of the column $H_{\text{col}} + 1$ , yielding factors of 3 or 4 relative to the original count, or involving subtraction terms.
Almost-Block Counting (Theorem 4.7): A crucial intermediate result is a formula for the number of packets containing the theta lift of an "almost-block." This extends previous results on blocks to handle the specific structures arising at the first occurrence.
Applications to Specific Representations:
- Generic Representations (Theorem 4.18): For generic representations of Arthur type, the paper determines that the number of containing packets is 1, 2, or 3. It explicitly demonstrates that non- $\theta$ -relevant pairs frequently exist even for generic representations.
- Unramified Representations (Theorem 4.20): For unramified representations of Arthur type, the count is shown to be either 2 or 3.
- Anti-Tempered Representations (Theorem 4.17): The paper extends the counting results to anti-tempered representations, providing a formula based on the dual of the representation.
Conjecture 4.12: The authors state a conjecture for the remaining seventh case (where the signs of the circles in consecutive columns are opposite). While not proven, Appendix A provides motivation for this formula.

Significance
The paper claims to provide a systematic investigation of non- $\theta$ -relevant pairs for the Adams conjecture. Its primary significance lies in:

Refuting the Sufficiency of the Adams Conjecture: It demonstrates that the theta lift of a tempered representation can lie in significantly more local Arthur packets than the single packet predicted by the Adams conjecture. The count can be 2, 3, or 4 times the original count, depending on the combinatorial structure of the representation.
Explicit Recursive Formulas: Unlike brute-force algorithmic approaches (which compute the lift and then the packet count separately), this work provides explicit recursive formulas that relate $|\Psi(\theta(\pi))|$ directly to $|\Psi(\pi)|$ and the structural parameters of $\pi$ . This allows for efficient computation without re-evaluating the entire theta correspondence.
Completing the Picture for Tempered Representations: By verifying six of the seven cases, the paper nearly completes the classification of the "failure" of the Adams conjecture for tempered representations at the first occurrence, bridging the gap between the known supercuspidal case and the general tempered case.

The authors note that their results rely heavily on the parameterization of local Arthur packets by extended multi-segments and the theory of their intersections, confirming that the "failure" of the Adams conjecture is a structured, combinatorial phenomenon rather than a random anomaly.

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