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Chebotarev geodesic theorem: split case

This paper generalizes prior work on the prime geodesic theorem in congruence classes of $\mathrm{SL}_2(\mathbb{Z})$ to prove that the geodesic analogue of the Chebotarev density theorem holds with an error exponent of $25/36 + \varepsilon$ , thereby establishing the same bound for the prime geodesic theorem across any congruence subgroup.

Original authors: Alberto Acosta Reche

Published 2026-06-25

📖 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 you are standing in a vast, infinite hall made of mirrors and curved walls (a mathematical space called the hyperbolic plane). In this hall, there are invisible "ghosts" that travel along the shortest possible paths, bouncing off the walls and eventually returning to their starting point. These paths are called geodesics.

Some of these paths are short loops, some are long, and some are incredibly long. Mathematicians are fascinated by counting these loops, much like a naturalist counting trees in a forest or stars in the sky. This counting problem is known as the Prime Geodesic Theorem.

The Main Goal: Counting the Ghosts

The paper asks a very specific question: If we count all these loops up to a certain length $X$ , how close is our count to the "expected" number?

In a perfect world, the number of loops would be exactly equal to $X$ . But in reality, there is always a little bit of "noise" or error. The paper is all about measuring how big this error is. The smaller the error, the better we understand the structure of the hall.

The New Discovery: A Better Map

Previous mathematicians had built maps to predict this error, but their maps were a bit blurry. They could say, "The error is roughly the size of $X$ raised to the power of 0.75" (which is still quite large).

The author of this paper, Alberto Acosta Reche, has drawn a much sharper map. He proves that for a specific type of hall (related to the group $SL_2(\mathbb{Z})$ and its variations), the error is actually much smaller—roughly the size of $X$ raised to the power of 25/36 (about 0.69).

Why does this matter?
Think of it like trying to guess the number of grains of sand on a beach.

Old Map: "I estimate there are 1,000,000 grains, give or take 100,000."
New Map: "I estimate there are 1,000,000 grains, give or take 10,000."
The new map gives a much more precise prediction.

The "Chebotarev" Twist: Sorting the Ghosts

The paper goes a step further. It doesn't just count all the loops; it sorts them into different "bins" based on their shape and how they twist around the hall. This is called the Chebotarev Geodesic Theorem.

Imagine you have a bag of mixed-up marbles (the loops). You want to sort them by color.

The Old Way: You could only count the total number of marbles.
The New Way: The author shows you can count the red marbles, the blue marbles, and the green marbles separately, and still get a very precise count for each color.

This is a big deal because it proves that the loops are distributed very evenly among these different "bins," just as prime numbers are distributed evenly among different remainders (a famous idea in number theory called the Chebotarev Density Theorem).

How Did They Do It? (The Toolkit)

To get this sharper map, the author had to use some very heavy-duty mathematical tools:

The Spectral Telescope: The author looks at the "vibrations" of the hall (like the sound of a bell). Every loop corresponds to a specific vibration. By studying these vibrations, he can count the loops without having to walk every single path.
The "Old" and "New" Forms: He had to organize these vibrations into families. Some vibrations are "new" (unique to this specific hall), and some are "old" (copies of vibrations from smaller, simpler halls). He figured out how to separate them so they wouldn't confuse the count.
The "Bykovskii-Zagier" Series: This is a special mathematical recipe (a type of infinite sum) that acts like a sieve. It helps filter out the noise and isolate the specific loops the author wants to count.

The "Split" Case

The title mentions the "split case." Imagine the hall is built from a specific type of brick (congruence subgroups). The author shows that no matter how you arrange these bricks (as long as they follow the rules), the counting rule holds true. This generalizes previous work that only worked for the simplest arrangement of bricks.

Summary

In simple terms, this paper is a major upgrade to the "GPS" mathematicians use to navigate the landscape of prime geodesics.

Before: We knew the general direction, but the error margin was wide.
Now: We have a much tighter error margin (25/36), and we can count the loops even when we sort them into specific categories.

This doesn't just solve a puzzle about loops in a mathematical hall; it strengthens the bridge between geometry (shapes and paths) and number theory (the study of numbers like primes), showing that the universe of numbers is more orderly and predictable than we previously thought.

Technical Summary of "Chebotarev geodesic theorem: split case" by Alberto Acosta Reche

Problem Statement
The paper investigates the Prime Geodesic Theorem (PGT) for congruence subgroups $\Gamma \subset SL_2(\mathbb{Z})$ , specifically focusing on the distribution of prime geodesics within specific conjugacy classes of the quotient group $\bar{\Gamma} = \Gamma' \backslash \Gamma$ , where $\Gamma'$ is a normal subgroup (typically a principal congruence subgroup $\Gamma(q)$ ). This problem is the geodesic analogue of the Chebotarev Density Theorem in algebraic number theory.

While the asymptotic behavior $\Psi_\Gamma(X) \sim X$ is well-established (with error terms $E_\Gamma(X)$ ), the precise rate of convergence for the error term in the general congruence case, particularly when weighted by non-trivial class functions, remains a central challenge. Previous work by Luo and Sarnak [LS95] and Soundararajan and Young [SY13] established strong bounds for the case $\Gamma = SL_2(\mathbb{Z})$ and specific subgroups, achieving an error term of $O_\epsilon(X^{25/36+\epsilon})$ . However, extending these results to arbitrary class functions on arbitrary congruence subgroups, and handling the technicalities of weight 1 Maass forms (arising when $-I \notin \Gamma(q)$ ), required further development.

Methodology
The author employs a combination of spectral theory, trace formulas, and analytic number theory techniques to generalize previous results. The methodology proceeds through several key stages:

Reduction to Congruence Characters: Using Artin's theorem on induced characters and properties of induced class functions, the problem of estimating sums over arbitrary class functions is reduced to estimating sums over congruence characters $\chi$ (Lemma 2.1, Proposition 2.3). This allows the focus to shift to the twisted Chebyshev sum $\Psi_\Gamma(X; \chi)$ .
Spectral Decomposition and Trace Formulas: The paper utilizes the Selberg trace formula and the Bruggeman–Kuznetsov trace formula. A significant technical hurdle addressed is the application of the Bruggeman–Kuznetsov formula to congruence characters where the test functions must satisfy specific growth conditions. The author adapts arguments from Proskurin [Pro03] and relaxes the conditions on test functions using Weil's bound on Kloosterman sums (Section 3.6).
Construction of Orthonormal Bases: A critical component is the construction of an orthonormal basis of Maass cusp forms for $(\Gamma, \chi)$ that interacts well with Hecke operators. The author proves the existence of a basis $\{u_j\}$ where each $u_j$ belongs to the sum of oldspaces of twists of a newform $f_j$ (Corollary 6.5). This structural result is essential for expressing the relevant spectral averages in terms of $L$ -functions.
Spectral Averages and $L$ -functions:
- Theorem 1.7 (Luo-Sarnak Generalization): The author generalizes the Luo-Sarnak argument to arbitrary congruence characters. This involves estimating the spectral average of the naive Rankin–Selberg series $L(s; |u_j|^2)$ . By decomposing the basis into oldspaces and utilizing the theory of newforms and twists, the author establishes a bound for the average of $|L(s; |u_j|^2)|^2$ on the critical line (Theorem 7.1). This relies on the spectral large sieve and bounds for adjoint $L$ -functions.
- Theorem 1.8 (Soundararajan-Young Generalization): The author generalizes the Soundararajan-Young approach using the Bykovskii–Zagier series $Z(s; t)$ . This series expresses the hyperbolic contribution to the trace formula in terms of Dirichlet $L$ -functions. The author proves a bound for $Z(s; t)$ on the critical line involving an admissible subconvexity exponent $\theta$ (Proposition 9.7), which relies on the factorization of $Z(s; t)$ into products of $L$ -functions and the definition of admissible exponents (Definition 1.1).
Optimization: The final error terms are derived by interpolating between the bounds obtained from the Luo-Sarnak strategy (Theorem 1.7) and the Soundararajan-Young strategy (Theorem 1.8), optimizing the parameters in the explicit formulas derived from the trace formulas.

Key Contributions and Results

Main Theorem (Theorem 1.2): The paper proves that for any congruence subgroup $\Gamma$ , any class function $f$ on $\bar{\Gamma}$ , and any admissible exponent $\theta > 0$ , the twisted Chebyshev sum satisfies:
$\Psi_\Gamma(X; f) = \langle f, 1_{\bar{\Gamma}} \rangle_{\bar{\Gamma}} X + O_{f, \theta, \epsilon}(X^{2/3 + \theta/6 + \epsilon}).$
Assuming the subconvexity exponent $\theta = 1/6 + \epsilon$ (which is known to be admissible by Petrow and Young [PY20, PY23]), the unconditional error term is $O_{f, \epsilon}(X^{25/36 + \epsilon})$ .
Corollary 1.3 (PGT for Congruence Subgroups): As a direct consequence, the Prime Geodesic Theorem holds for any congruence subgroup $\Gamma$ with the error term $E_\Gamma(X) = O_\epsilon(X^{25/36 + \epsilon})$ . This strengthens the claim in [LRS95] that the Luo-Sarnak bound could be generalized.
Corollary 1.4 (Geodesic Dirichlet Theorem): The paper establishes a geodesic analogue of Dirichlet's theorem on arithmetic progressions. For any $a \in \mathbb{Z}/q\mathbb{Z}$ , the sum of geodesics with trace congruent to $a$ modulo $q$ is asymptotically proportional to the size of the fiber of the trace map, with the same error term $O(X^{25/36+\epsilon})$ . This improves upon previous results in [CHK24] and [Dei25] regarding the error term in the $X$ -aspect.
Theorems 1.7 and 1.8: These intermediate results provide the necessary bounds for the exponential sums $S(T, X)$ over spectral parameters. Theorem 1.7 generalizes the Luo-Sarnak bound ( $S(T, X) \ll T^{5/4}X^{1/8}$ ), and Theorem 1.8 generalizes the Soundararajan-Young bound ( $S(T, X) \ll X^{1/4+\theta/2}T^{1/2}$ ).

Significance and Claims
The paper claims to provide the first unconditional proof that the Chebotarev geodesic theorem holds with the exponent $25/36 + \epsilon$ for arbitrary congruence subgroups and arbitrary class functions. This unifies and extends the results of Luo, Sarnak, Soundararajan, and Young from the specific case of $SL_2(\mathbb{Z})$ and constant functions to the general setting of congruence characters.

The author explicitly notes that the setup is slightly more general than Sarnak's original work [Sar80] because it accommodates congruence subgroups that do not contain $-I$ (requiring the treatment of Maass forms of weight 1). The significance lies in the successful adaptation of the "split case" arguments (using the Bykovskii–Zagier series and Rankin–Selberg convolutions) to this broader context, thereby confirming the expected behavior of the error term for a wide class of hyperbolic manifolds. The paper does not claim to reach the conjectured optimal bound of $X^{1/2+\epsilon}$ , noting that such a bound remains far out of reach, but it establishes the current best-known unconditional bound for this general setting.