Effective correlation and decorrelation for newforms, and weak subconvexity for $L$-functions

1. Problem Statement

The paper addresses two interconnected problems in analytic number theory and the spectral theory of automorphic forms:

1. Effective Quantum Unique Ergodicity (QUE): The classical QUE conjecture (Lindenstrauss, Soundararajan, etc.) posits that eigenfunctions of the Laplacian on a hyperbolic surface become equidistributed as the eigenvalue (or weight $k$) tends to infinity. While the qualitative limit is known, obtaining effective rates of convergence (how fast the equidistribution occurs) with explicit dependence on the level $q$, weight $k$, and the test function $\psi$ remains a challenge. Specifically, the authors aim to bound the inner product:
   $$\langle \psi, |F_k|^2 \rangle_q - \frac{3}{\pi}\langle \psi, 1 \rangle_1$$
   where $F_k$ is a normalized holomorphic newform of weight $k$ and level $q$.

2. Effective Decorrelation: When considering two distinct newforms $f$ and $g$ (with $f \neq g$), the paper investigates the decay of their correlation:
   $$\langle \psi, F_k G_k \rangle_q$$
   The goal is to show this tends to zero effectively as $kq \to \infty$, extending previous results (like Huang's for level $q=1$) to general squarefree levels.

2. Methodology

The authors employ a hybrid approach combining two distinct methods for bounding spectral sums, refined by new analytic estimates on $L$-functions.

A. Refinement of Weak Subconvexity (Soundararajan's Method)

The core analytic innovation is a refinement of Soundararajan's weak subconvexity bound for Rankin–Selberg $L$-functions $L(s, \pi \times \pi')$.

• The Bound: The authors prove a bound of the form:
  $$L(1/2, \pi \times \pi') \ll_{m,m',\varepsilon} C(\pi')^\varepsilon \frac{C(\pi \times \pi')^{1/4}}{(\log C(\pi \times \pi'))^{1-\varepsilon}}$$
  Crucially, this bound is uniform in the conductor of the second representation $\pi'$ (up to a small power $C(\pi')^\varepsilon$) and saves a power of the logarithm close to 1.
• Technique: The proof follows Soundararajan's framework using the "slow variation of mean values" of multiplicative functions. It involves:
  • Constructing a smoothed sum $S(x)$ using a specific kernel involving parameters $\tau_j$.
  • Using an approximate functional equation.
  • Establishing "successive maxima" of the $L$-function on the critical line to control the error terms.
  • Utilizing bounds for partial sums of Dirichlet coefficients (Lemma 3.5, 3.7) derived from the Generalized Ramanujan Conjecture (GRC).

B. Holowinsky's Method (Shifted Convolution Sums)

The second approach relies on expanding the test function $\psi$ into incomplete Poincaré series and Eisenstein series.

• Unfolding: The inner product is unfolded to express it as sums involving shifted convolution sums of Hecke eigenvalues: $\sum \lambda_f(n)\lambda_g(n+m)$.
• Optimization: The authors combine the Soundararajan bound (effective for small shifts) and the Holowinsky bound (effective for large shifts) using a minimization argument (optimizing a parameter $\alpha \in [0,1]$) to derive the final rate.

C. Spectral Decomposition

The error term in the QUE/decorrelation problem is decomposed via the spectral expansion of $L^2(Y_0(q))$:
$$\langle \psi, F_k G_k \rangle_q - \text{Main Term} = \sum_j \langle \psi, \phi_j \rangle \langle \phi_j, F_k G_k \rangle + \int \langle \psi, E_t \rangle \langle E_t, F_k G_k \rangle dt$$
The authors bound the triple product integrals $\langle \phi_j, F_k G_k \rangle$ and $\langle E_t, F_k G_k \rangle$ using Watson's triple product formula (for Maaß forms) and Rankin–Selberg theory, relying heavily on the new subconvexity bounds.

3. Key Contributions

1. Theorem 1.3 (Refined Weak Subconvexity): A new, stronger bound for $L(1/2, \pi \times \pi')$ that is uniform in the conductor of $\pi'$ with a logarithmic saving close to 1. This is the engine driving the effective results.
2. Correction of Previous Literature (Appendix A): The appendix by Jesse Thorner identifies a missing factor in the proof of Iwaniec's course notes and the subsequent work by Lester, Matomäki, and Radziwiłł regarding the QUE rate for level 1. They show the convergence rate is slightly slower than previously claimed and provide a corrected exponent.
3. Effective Bounds for General Levels: The paper extends effective decorrelation results from level $q=1$ to arbitrary squarefree levels $q$, and provides effective QUE bounds for varying levels.

4. Main Results

Theorem 1.1 (Main Theorem):
Let $f, g$ be holomorphic newforms of weight $k$ and level $q$. Let $q_0$ be the largest squarefree divisor of $q$. If $f \neq g$, assume $q$ is squarefree. For a test function $\psi \in C_c^\infty(Y_0(1), M)$, the following bound holds:
$$\left| \langle \psi, F_k G_k \rangle_q - \mathbb{1}_{f=g}\frac{3}{\pi}\langle \psi, 1 \rangle_1 \right| \ll_{\varepsilon} M^4 \left(\frac{q}{q_0}\right)^{-\frac{1}{12}(2\sqrt{3}-3)(1-2\theta)+\varepsilon} (\log kq)^{-(\frac{7}{2}-2\sqrt{3})+\varepsilon}$$
where $\theta \le 7/64$ is the bound towards the Ramanujan conjecture.

• Case $f=g$ (QUE): This yields an effective rate of convergence for the equidistribution of mass. The exponent of the logarithm is $\delta = \frac{7}{2} - 2\sqrt{3} \approx 0.0359$.
• Case $f \neq g$ (Decorrelation): This establishes that distinct newforms decorrelate effectively, extending Huang's result to squarefree levels.

Corollary 1.2 (Equidistribution of Zeros):
Using the improved QUE rate, the authors deduce that the zeros of $f$ equidistribute on the fundamental domain with a specific effective rate. Specifically, for a hyperbolic ball of radius $r \ge (\log k)^{-1/203}$, the proportion of zeros in the ball converges to the volume measure with an error term of $O((\log k)^{-\delta})$.

Refinement for Level 1:
When $q=1$ and $f=g$, the authors improve the dependence on the test function parameter $M$ from $M^4$ to $M^{(23-2\sqrt{3})/12} \approx M^{1.63}$.

5. Significance

• Quantitative Progress: The paper moves beyond qualitative "limit" theorems to provide explicit, effective rates of convergence for QUE and decorrelation. This is crucial for applications where the level or weight is large but finite.
• Uniformity: The results are uniform in the level $q$ (specifically handling the squarefree part $q_0$) and the weight $k$, which is a significant technical achievement in the theory of automorphic forms.
• Methodological Improvement: The refinement of the weak subconvexity bound (Theorem 1.3) is likely to be applicable to other problems involving Rankin–Selberg $L$-functions where uniformity in the conductor is required.
• Correction of Error: The appendix corrects a subtle but important error in the literature regarding the convergence rate of zeros for level 1 forms, ensuring the mathematical record is accurate.
• Optimization: The paper demonstrates the power of combining the "spectral" (Soundararajan) and "arithmetic" (Holowinsky) approaches, optimizing the trade-off between them to achieve the best known unconditional bounds.

In summary, this work represents a significant step forward in the quantitative understanding of the distribution of mass and zeros of automorphic forms, providing the sharpest known unconditional effective bounds for these phenomena.