The Second Moment of Rankin-Selberg $L$-Functions in Conductor-Dropping Regimes

Technical Summary: The Second Moment of Rankin–Selberg L-Functions in Conductor-Dropping Regimes

Problem Statement
This paper addresses the asymptotic behavior of the second moment of Rankin–Selberg $L$-functions associated with the convolution of two holomorphic Hecke cusp forms, $f$ and $g$, of equal weight $k$. Specifically, the authors investigate the sum:
$$\sum_{f \in B_{\text{hol}}^{k}} \frac{L(1/2, f \otimes g)^2}{L(1, \text{ad } f)}$$
where $B_{\text{hol}}^{k}$ is an orthonormal basis of holomorphic Hecke cusp forms of weight $k$.

This problem is situated within the broader context of the subconvexity problem for $L$-functions. A primary motivation is the adjoint $L$-function $L(s, \text{ad } f)$, where $g=f$. The central value $L(1/2, \text{ad } f)$ is bounded by the convexity bound $k^{1/2}$, and obtaining subconvex bounds (saving a power of $k$) is a major open problem. Standard heuristics suggest that an asymptotic formula for the second moment with a power-saving error term could facilitate subconvexity results via amplification. However, in "conductor-dropping" regimes—where the analytic conductor of $L(1/2, f \otimes g)$ drops from $k^4$ to $k^2$ because the weights are equal—standard methods often fail to produce asymptotic formulas with power-saving errors. While upper bounds of the form $O_\epsilon(k^{1+\epsilon})$ are known (via large sieve inequalities), deriving a precise asymptotic formula with a main term and a controlled error term has remained elusive.

Methodology
The authors employ a spectral expansion technique involving the period integral of four automorphic forms. The core strategy involves evaluating the integral
$$\int_{SL_2(\mathbb{Z})\backslash \mathbb{H}} |G(z)|^2 E^*(z, s_1) E^*(z, s_2) \, d\mu(z)$$
where $G(z) = y^{k/2}g(z)$ and $E^*(z, s)$ is the completed real-analytic Eisenstein series. This integral is expanded spectrally in two distinct ways:

1. First Spectral Expansion (Weight $k$): The integral is viewed as the inner product $\langle G E^*(\cdot, s_1), G E^*(\cdot, s_2) \rangle$. Using Parseval's identity for square-integrable automorphic forms of weight $k$, the authors expand this in terms of holomorphic cusp forms, Maass cusp forms, and Eisenstein series. This yields a sum involving the target moment $M_{\text{hol}}$ (the sum over weight $k$ forms) and other spectral terms ($M_{\text{Maass}}$, $M_{\text{Eis}}$). The inner products are computed explicitly using the Rankin–Selberg method, expressing them in terms of $L$-functions and Gamma factors.

2. Second Spectral Expansion (Weight 0): The integral is viewed as the inner product $\langle E^*(\cdot, s_1) E^*(\cdot, s_2), |G|^2 \rangle$. Since Eisenstein series are not square-integrable, the authors utilize a regularized version of Parseval's identity (following Zagier). This expansion involves the continuous spectrum and discrete spectrum of weight 0 forms. Crucially, the inner products involving $|G|^2$ and Maass forms are evaluated using the Watson–Ichino triple product formula, which relates these integrals to central values of triple product $L$-functions.

By equating the results of these two expansions, the authors derive an exact identity relating the target moment to a "main term" (arising from the continuous spectrum and residues) and various error terms involving spectral sums.

Key Contributions and Results
The paper establishes the following main theorem (Theorem 1.4):

• Asymptotic Formula: For a fixed holomorphic Hecke cusp form $g$ of weight $k$, the second moment satisfies:
  $$\sum_{f \in B_{\text{hol}}^{k}} \frac{L(1/2, f \otimes g)^2}{L(1, \text{ad } f)} = k \sum_{j=0}^3 P_j(\log k) L^{(3-j)}(1, \text{ad } g) + O_\epsilon(k (\log k)^{-1/2 + \epsilon})$$
  where $P_j$ are polynomials of degree $j$.
• Main Term Analysis: The main term is shown to be positive for sufficiently large $k$ and satisfies the bounds $k(\log k)^2 \ll \text{Main Term} \ll k(\log k)^6$.
• Error Term: The error term achieves a logarithmic saving, $O_\epsilon(k (\log k)^{-1/2 + \epsilon})$, rather than a power saving.

Technical Challenges and Solutions
The authors highlight several specific difficulties overcome in this work:

1. Lack of Power-Saving Error: The authors demonstrate that obtaining a power-saving error term in this conductor-dropping regime is likely impossible without assuming pre-existing subconvex bounds for $L(1/2, \text{ad } g)$ and its $GL_2$ twists. Consequently, the proof relies on Soundararajan's weak subconvex bounds to achieve the logarithmic saving.
2. Positivity of the Main Term: A significant technical hurdle is proving that the main term dominates the error term. The main term involves $L(1, \text{ad } g)$ and its derivatives, for which only weak unconditional bounds are known ($L(1, \text{ad } g) \gg (\log k)^{-1}$). The authors circumvent the potential cancellation between diagonal and off-diagonal terms by exploiting a "hidden structure." They express the main term as an integral that can be compared term-by-term with a Dirichlet series representation, showing that the $n=1$ term (which is $\gg k (\log k)^2$) dominates the sum.
3. Generalizations: The methods are shown to extend to:
   • The level aspect (varying prime level $q$ with fixed weight).
   • The spectral aspect (averaging over Maass forms with spectral parameter in $[T, T+1]$).
   • Shifted central points ($1/2 + it$).
   • The first moment of triple product $L$-functions (though proving the main term dominates the error term in this case remains an open difficulty due to lower bound issues).

Significance and Claims
The paper claims to provide the first asymptotic formula for this specific second moment in the conductor-dropping regime. While the error term does not save a power of $k$, the result is significant because:

• It confirms the existence of an asymptotic formula where only upper bounds were previously known.
• It refines the understanding of moments in conductor-dropping settings, where standard heuristics can be misleading.
• It improves upon previous upper bound results by Blomer by providing a precise main term and a spectral expansion approach.
• The authors explicitly state that their method relies on the Watson–Ichino formula, which limits its applicability to $GL_2$ (as such a formula does not exist for $GL_n$ with $n \ge 3$), distinguishing their approach from Blomer's more general upper bound techniques.

The paper concludes that while the result does not immediately yield subconvexity bounds (due to the lack of power-saving error), it represents a necessary step in understanding the distribution of these $L$-values and the structural constraints of conductor-dropping moments.