Prime geodesic theorem for the Picard manifold

Abstract

Let $\mathrm{\Gamma }=\mathbf{PSL}(2,\mathbf{Z}[i])$ be the Picard group and ${\mathbf{H}}^3$ be the three-dimensional hyperbolic space. We study the Prime Geodesic Theorem for the quotient $\mathrm{\Gamma }\setminus {\mathbf{H}}^3$, called the Picard manifold, obtaining an error term of size $O(X^{3/2+\theta /2+ϵ})$, where θ denotes a subconvexity exponent for quadratic Dirichlet L-functions defined over Gaussian integers.

Introduction

The classical Prime Geodesic Theorem states that the counting function $\pi (X)$ of primitive hyperbolic classes in ${\mathbf{PSL}}_2(\mathbf{Z})$ whose norm does not exceed X satisfies the asymptotic law

$$\pi (X)\sim \mathrm{Li}(X)\text{ as }X\to \infty ,$$

where $\mathrm{Li}(X)$ is the logarithmic integral. This theorem and its generalisations can be considered as geometric analogues of the Prime Number Theorem, while norms of primitive hyperbolic elements are sometimes called “pseudoprimes”.

In this paper we study the three dimensional version of the Prime Geodesic Theorem. Different concepts of the two dimensional theory can be extended to this case in a natural and elegant way. The role of the Poincaré upper half plane ${\mathbf{H}}^2$ is now played by the three dimensional hyperbolic space

$${\mathbf{H}}^3=\{(z,r);z=x+iy\in \mathbf{C};r>0\}.$$

Let $\mathrm{\Gamma }\subset \mathbf{PSL}(2,\mathbf{C})$ be a discrete cofinite group. Prime Geodesic Theorem for the hyperbolic manifold $\mathrm{\Gamma }\setminus {\mathbf{H}}^3$ provides an asymptotic formula for the function ${\pi }_{\mathrm{\Gamma }}(X)$, which counts the number of primitive hyperbolic or loxodromic elements in Γ with norm less than or equal to X.

In the pioneering paper [26], Sarnak proved the following asymptotic formula

$${\pi }_{\mathrm{\Gamma }}(X)=\mathrm{Li}(X^2)+O(X^{3/2+1/6+ϵ}).$$

Further progress has been made in the case of the Picard group $\mathrm{\Gamma }=\mathbf{PSL}(2,\mathbf{Z}[i])$ defined over Gaussian integers

$$\mathbf{Z}[i]=\{a+bi;a,b\in \mathbf{Z}\}.$$

Assuming the mean Lindelöf hypothesis for symmetric square L-functions attached to Maass forms on the Picard manifold $\mathrm{\Gamma }\setminus {\mathbf{H}}^3$, Koyama [14] obtained an error term of size

$$O(X^{3/2+1/14+ϵ}).$$

Finally, an unconditional improvement of Sarnak's estimate (1.2), namely

$$O(X^{3/2+1/8+ϵ}),$$

was derived in [1] as a consequence of a non-trivial estimate for the second moment of symmetric-square L-functions.

Proofs of the last two results are centered around Nakasuji's explicit formula proved in [20, Theorem 4.1] and [21, Theorem 5.2]. Let us denote the remainder term in the prime geodesic theorem by $E_{\mathrm{\Gamma }}(X)$. Nakasuji showed that for $1\le T\le X^{1/2}$ we have

$$E_{\mathrm{\Gamma }}(X)=2\Re \left(\sum _{0<r_j\le T}\frac{X^{1+i{r}_j}}{1+i{r}_j}\right)+O(\frac{X^2}{T}\log X),$$

where ${\lambda }_j=1+r_j^2$ are the eigenvalues of the hyperbolic Laplacian on $L^2(\mathrm{\Gamma }\setminus {\mathbf{H}}^3)$. Explicit formula (1.5) brings into play the spectral exponential sum

$$S(T,X)=\sum _{0<r_j\le T}{X}^{i{r}_j}.$$

The trivial bound

$$S(T,X)\ll T^3$$

follows from Weyl's law and yields Sarnak's result (1.2).

The proof of (1.4) is based on the following improvement (see [1, Theorem 3.2])

$$S(T,X)\ll T^2{X}^{1/4}{(TX)}^{ϵ}.$$

The aim of this paper is to obtain a new upper bound for the spectral exponential sum, which explicitly depends on a subconvexity exponent θ for quadratic Dirichlet L-functions defined over Gaussian integers.

Theorem 1.1

For $1\le T\le X^{1/2}$ the following estimate holds

$$S(T,X)\ll X^{1/2+\theta /2}T{(TX)}^{ϵ}.$$

As discussed in [1, Remark 3.1], it is not obvious what is the correct order of magnitude of $S(T,X)$ for all X and T. In the two dimensional case, Petridis and Risager conjectured in [24] that the spectral exponential sum exhibits square root cancellation, and Laaksonen verified this conjecture numerically.

The most important consequence of Theorem 1.1 is the new estimate on $E_{\mathrm{\Gamma }}(X)$.

Theorem 1.2

The error term in the Prime Geodesic Theorem for the Picard manifold can be estimated as follows

$$E_{\mathit{\Gamma }}(X)\ll X^{3/2+\theta /2+ϵ}.$$

Assuming the Lindelöf hypothesis $\theta =0$, we improve the conditional result (1.3) and establish for the first time the error estimate $O(X^{3/2+ϵ})$. Note that this is the best possible error admissible by the explicit formula of Nakasuji (1.5). Recently, the exponent 3/2 was improved conditionally by Balog-Biro-Cherubini-Laaksonen [5] under two standard conjectures for L-functions defined over Gaussian integers: the Lindelöf hypothesis for quadratic Dirichlet L-functions and the mean Lindelöf hypothesis for the second moment of symmetric square L-functions. Furthermore, the authors of [5] established a new unconditional estimate, namely $O(X^{3/2+2/21+ϵ})$.

Theorem 1.1 is a consequence of the following estimate for the mean value of Maaß Rankin-Selberg L-functions on the critical line multiplied by the oscillating factor $X^{i{r}_j}$.

Theorem 1.3

Let $X\gg 1$ and $X^{ϵ}\le T\le X^{1/2}$. Then for $s=1/2+it,|t|\ll T^{ϵ}$ we have

$$\sum _{r_j}\frac{r_j}{\sinh (\pi r_j)}{\omega }_T(r_j)X^{i{r}_j}L(u_j\otimes u_j,s)\ll T^{3/2}{X}^{1/2+\theta +ϵ},$$

where ${\omega }_T(r_j)$ is a smooth characteristic function of the interval $(T,2T)$.

The standard method for investigating the left-hand side of (1.10) is to estimate everything by absolute value so that $X^{i{r}_j}$ is simply replaced by 1 and the modulus of the Rankin-Selberg L-function is bounded by its square by the means of the Cauchy-Schwarz inequality. In this way, the problem is reduced to studying the second moment. This technique allows proving non-trivial results in certain ranges but the disadvantage is that the information about the behaviour of $X^{i{r}_j}$ is completely lost.

The main novelty and the core idea of our approach is that absolute value estimates are replaced by the method of analytic continuation. More precisely, we prove an exact formula for the left-hand side of (1.10) which allows us to take into consideration oscillations of the weight function $X^{i{r}_j}$. This approach has already proved to be effective in the two dimensional case (see [3]), motivating us to develop the method further and to study the Prime Geodesic Theorem for the Picard manifold. We remark that in the three-dimensional setting, new technical difficulties arise (i.e. more complicated special functions in the Kuznetsov trace formula), requiring the change of methodology and more involved analysis.