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Chebotarev geodesic theorem: non-split case

This paper establishes the prime geodesic theorem with an exponent of $25/36 + \varepsilon$ for congruence subgroups of indefinite quaternion orders over $\mathbb{Q}$ , including the non-split case, by reducing the problem to the previously solved split case.

Original authors: Alberto Acosta Reche

Published 2026-08-06

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Alberto Acosta Reche

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 universe of numbers as a vast, invisible landscape where prime numbers are the most fundamental landmarks. For centuries, mathematicians have been trying to map this terrain, asking how these prime numbers are scattered across the number line. It's a bit like trying to predict where the next star will appear in the night sky, but instead of stars, we are looking for the building blocks of arithmetic. To do this, they often use a tool called a "geodesic," which is just a fancy word for the shortest path between two points on a curved surface, like a great circle route on a globe. In the world of number theory, these paths are loops that wrap around a strange, hyperbolic shape. The "Prime Geodesic Theorem" is the rulebook that tells us how many of these loops exist as we look further and further out.

Now, imagine you are not just looking at a simple globe, but a complex, multi-layered maze made of different kinds of mathematical materials. Some parts of this maze are "split," meaning they are easy to unfold and understand, while others are "non-split," meaning they are twisted and knotted in ways that make them much harder to navigate. For a long time, mathematicians could only accurately count the paths in the easy, split sections. The hard, twisted sections were a mystery, and the estimates for how many paths they contained were a bit fuzzy, leaving a large margin of error. This paper steps into that twisted, non-split maze to see if we can finally get a clear count of the paths, just like we can in the easy sections.

The author, Alberto Acosta Reche, tackles a specific type of mathematical maze built from "indefinite quaternion orders." Think of these as a special kind of 4-dimensional puzzle piece that, when stretched out, looks like a 2D surface with a lot of curvature. The goal is to count the "prime geodesics"—the shortest, non-repeating loops—on these surfaces. The paper proves that for these tricky, non-split mazes, we can now count these loops with a very high degree of precision. Specifically, the error in the count (the difference between the actual number and the predicted number) is no bigger than a specific mathematical limit: $X^{25/36+\varepsilon}$ .

To put this in perspective, imagine you are counting the number of grains of sand on a beach that grows as you walk. If you guess the number, you might be off by a few grains, or maybe a few thousand. The "error term" is how many grains you might be wrong by. Before this paper, for these specific twisted mazes, the guess could be off by a huge amount. This paper shows that the guess is actually much, much closer to the truth. The author achieves this by using a clever trick: instead of trying to solve the twisted maze directly, they translate the problem into the language of the "split" mazes, which are already well-understood. They show that the twisted paths behave in a way that is mathematically identical to paths in the split mazes, just with a slightly different level of complexity.

The paper doesn't just guess; it provides a rigorous proof. It establishes that the "Chebotarev geodesic theorem"—a rule that predicts how these paths are distributed among different types of loops—holds true for these quaternion groups with the same high precision as the simpler groups. The author proves that the error term is bounded by $X^{25/36+\varepsilon}$ , where $X$ is the size of the area you are looking at, and $\varepsilon$ is a tiny, flexible number that can be made as small as needed. This result is significant because it extends the best-known accuracy for counting these paths from the simple, split cases to a much wider and more complex class of mathematical groups. It's like finally getting a high-definition map for a region of the world that was previously only visible through a foggy window, confirming that the rules of the universe hold up even in the most twisted corners of the mathematical landscape.

Technical Summary: Chebotarev Geodesic Theorem: Non-Split Case

Problem Statement
The paper addresses the Prime Geodesic Theorem (PGT) for congruence subgroups of indefinite quaternion orders over $\mathbb{Q}$ . Specifically, it investigates the asymptotic behavior of the weighted sum of lengths of closed geodesics on the hyperbolic orbifold associated with these groups. The central problem is to establish the geodesic analogue of the Chebotarev density theorem (CGT) for these "non-split" (division algebra) cases with a sharp error term.

While the PGT and CGT for the split case ( $\Gamma = \text{SL}_2(\mathbb{Z})$ ) and its congruence subgroups have been extensively studied, with the best known error term for the split case being $O_\varepsilon(X^{25/36+\varepsilon})$ (Soundararajan and Young), the non-split case (indefinite quaternion division algebras) had previously yielded weaker bounds (e.g., exponent $7/10 + \varepsilon$ for principal congruence subgroups). The goal is to extend the $25/36 + \varepsilon$ exponent to the non-split setting.

Methodology
The proof strategy relies on reducing the non-split problem to the split case through the explicit matching of orbital integrals. The methodology proceeds in three main stages:

Reduction to Stable Functions:
The author first addresses the issue that the test functions $\phi$ on the finite quotient $(O/qO)^1$ may not be "stable" (invariant under conjugation by the unit group). Using Artin reciprocity for quadratic field extensions, the paper proves that averaging a function over the conjugation action introduces only a small error term ( $O(X^{1/2+\varepsilon})$ ). This allows the assumption that the test function is stable without significantly affecting the asymptotic count.
Local Orbital Integral Matching:
The core technical contribution involves constructing a matching between orbital integrals on the division algebra side ( $D^\times$ ) and the split algebra side ( $\text{GL}_2$ ).
- The author defines "matching functions" $\phi$ on $D^\times$ and $f$ on $\text{GL}_2(\mathbb{Q}_p)$ such that their orbital integrals coincide for all relevant quadratic algebras $E$ and elements $x$ .
- Unlike previous versions of the Jacquet–Langlands correspondence (e.g., in [GJ79]), this work provides an explicit construction that tracks the support and level of the functions.
- Specifically, for a function $\phi$ of level $p^n$ on the maximal order of the division algebra, the author constructs a matching function $f$ of level $p^{n+1}$ on $\text{GL}_2(\mathbb{Z}_p)$ (or $\text{SL}_2(\mathbb{Z}_p)$ ). This construction distinguishes between cases where the element's "depth" is even or odd, utilizing specific hereditary orders ( $M$ , $J$ , and the maximal order of $D$ ) to define the matching.
Global Synthesis:
By combining the local matching results with the adelic formulation of the geodesic sums, the author establishes a global identity. For a stable class function $\phi$ on a congruence subgroup of an indefinite quaternion division algebra $D$ , there exists a corresponding stable class function $f$ on $\text{SL}_2(\mathbb{Z}/q'\mathbb{Z})$ (where $q' = \text{disc}(D)q$ ) such that the geodesic sums are identical up to a negligible error:
$\Psi_{O^1}(X; \phi) = \Psi_{\text{SL}_2(\mathbb{Z})}(X; f) + O_{\varepsilon, \phi}(X^{1/2+\varepsilon})$

Key Results
The paper establishes the following main theorem and corollaries:

Theorem 1.2: For any congruence subgroup $\Gamma$ of level $q$ in an indefinite quaternion order $O^1$ , and any class function $\phi$ , the geodesic sum $\Psi_\Gamma(X; \phi)$ equals the corresponding split sum $\Psi_{\text{SL}_2(\mathbb{Z})}(X; f)$ plus an error term of $O(X^{1/2+\varepsilon})$ .
Corollary 1.3 (Chebotarev Geodesic Theorem): Combining Theorem 1.2 with the best known bounds for the split case, the paper proves that for any admissible exponent $\theta > 0$ (where $\theta = 1/6 + \varepsilon$ is currently admissible), the error term for the twisted sum is:
$\Psi_\Gamma(X; \phi) = \langle \phi, 1 \rangle X + O(X^{2/3 + \theta/6 + \varepsilon})$
Corollary 1.4 (Prime Geodesic Theorem): By setting $\phi = 1$ , the paper derives the unconditional bound for the standard PGT in the non-split case:
$\Psi_\Gamma(X) = X + O_\Gamma(X^{25/36+\varepsilon})$
This matches the current strongest known result for $\text{SL}_2(\mathbb{Z})$ .

Significance and Claims
The paper claims to extend the current strongest form of the Prime Geodesic Theorem from the split case ( $\text{SL}_2(\mathbb{Z})$ ) to a much wider class of groups: congruence subgroups of indefinite quaternion orders.

Improvement over Prior Work: The result improves upon the exponent $7/10 + \varepsilon$ found in [Tan+26] (which applied only to principal congruence subgroups) to $25/36 + \varepsilon$ for arbitrary congruence subgroups.
Explicit Matching: The paper provides a completely explicit version of the matching of local orbital integrals for inner forms of $\text{GL}_2$ , specifically tracking the level and support of the test functions, which was a limitation in previous literature.
Methodological Independence: The proof of the stability reduction (Proposition 1.7) is achieved "by hand" using Artin reciprocity, avoiding reliance on the stable trace formula or deep background results on the Arthur trace formula.

The work effectively demonstrates that the non-split case can be reduced to the split case with high precision, thereby transferring the best known analytic number theory bounds from the modular surface to quaternionic orbifolds.

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