Second moment of $\textrm{GL(3)} \times \textrm{GL(2)}$ $L$--functions

1. Problem Statement and Context

The Core Problem:
The paper investigates the second moment of Rankin-Selberg $L$-functions associated with the pair $(\pi, f)$, where $\pi$ is a fixed $SL(3, \mathbb{Z})$ Hecke cusp form and $f$ varies over a family of primitive newforms of $GL(2)$. Specifically, the authors aim to establish a Lindelöf-consistent upper bound for the sum:
$$S_\pi(M) = \sum_{\substack{\psi \pmod{M_1} \\ \psi(-1)=(-1)^k}} \sum_{f \in H^*_k(M_1 M_2, \psi)} |L(1/2, \pi \times f)|^2$$
where:

• $M_1$ and $M_2$ are distinct primes.
• The level of the newforms is $M = M_1 M_2$.
• The weight is $k \geq 3$.
• The nebentypus $\psi$ has conductor $M_1$.

The Goal:
To prove that $S_\pi(M) \ll_{\pi, \epsilon} M_1^{1+\epsilon}$ in the range $M_2 \leq M_1^{1+\epsilon}$.
This result is significant because it provides a "non-archimedean" analog to recent results by Qi (2026) regarding short spectral intervals for $GL(2)$ forms, extending the understanding of moments in higher-rank families ($GL(3) \times GL(2)$).

Motivation:
While asymptotic formulas for moments of $L$-functions are known for low-rank families (like $GL(1)$ and $GL(2)$) in specific ranges, obtaining such bounds for higher-rank families (like $GL(3) \times GL(2)$) is notoriously difficult. The Lindelöf hypothesis suggests the bound should be roughly the size of the family ($|Family| \approx M_1^2 M_2$), but proving even the weaker "Lindelöf-consistent" bound ($|Family|^{1+\epsilon}$) is a major challenge in analytic number theory.

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2. Methodology

The proof employs a sophisticated combination of analytic tools, primarily centered around the Petersson Trace Formula, Voronoi Summation, and Large Sieve Inequalities.

A. Approximate Functional Equation and Petersson Trace Formula

1. Approximate Functional Equation: The authors first express $|L(1/2, \pi \times f)|^2$ as a sum involving the Fourier coefficients of $\pi$ and $f$, weighted by a smooth function $W$.
2. Petersson Trace Formula: They apply the Petersson trace formula to the sum over the family of newforms $f$. This transforms the problem from estimating a sum of $L$-values to estimating a sum of Kloosterman sums and Bessel functions.
   • The diagonal term yields the main contribution.
   • The off-diagonal term (denoted $O_D$) involves a sum over moduli $c$ and Kloosterman sums $S_\psi(m, m'; Mc)$.

B. Separation of Variables

The off-diagonal term is complex because the variables $m$ and $m'$ are entangled in the Kloosterman sums and the Bessel function argument. The authors use two key techniques to separate them:

1. Character Sum Decomposition (Lemma 8): Using Lemma 5 (a reciprocity-like identity for character sums), they decompose the character sum $C_{Mc}(m, m')$ into products of simpler exponential sums involving variables $m$ and $m'$ separately.
2. Mellin Inversion: To handle the Bessel function term $J_{k-1}(4\pi\sqrt{mm'}/Mc)$, they use a Mellin inversion trick. This introduces a dual variable $y$ and allows the separation of $m$ and $m'$ in the weight function, converting the problem into an integral over $y$.

C. GL(3) Voronoi Summation

After separating variables, the sum over $m$ (and $m'$) involves the Fourier coefficients of the $GL(3)$ form $\pi$, denoted $\lambda_\pi(\ell, m)$.

• The authors apply the $GL(3)$ Voronoi summation formula (Lemma 6) to these sums.
• This transforms the sum over $m$ into a dual sum involving Kloosterman sums and a new integral transform $G_\pm(x)$.
• The analysis of $G_\pm(x)$ is split into "small" and "large" arguments, utilizing stationary phase methods and integration by parts to bound the oscillatory integrals.

D. Large Sieve Inequalities

The core of the analytic difficulty lies in bounding the resulting sums over the dual variables. The authors prove and utilize a Hybrid Large Sieve Inequality (Lemma 4).

• This lemma bounds sums of the form $\sum | \sum a_m e(tf(m)) e(\gamma m / cD) |^2$.
• It combines the standard large sieve (for arithmetic progressions) with bounds for exponential sums with smooth phases (handling the $GL(3)$ oscillatory terms).
• This allows them to control the size of the sums over the moduli $c$ and the spectral parameter $y$.

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3. Key Contributions

1. Extension of Qi's Result: The paper successfully extends the "non-archimedean" analog of Qi's short-interval results from $GL(2)$ to the $GL(3) \times GL(2)$ setting.
2. New Large Sieve Bound: The authors prove a specific hybrid large sieve inequality (Lemma 4) tailored to handle the oscillatory nature of $GL(3)$ coefficients combined with arithmetic characters. This is a critical technical ingredient for the proof.
3. Handling the "Short Family" Regime: The result holds in the range $M_2 \leq M_1^{1+\epsilon}$. The authors note that the case $M_2 \geq M_1^{1+\epsilon}$ is currently out of reach because the family size becomes too small relative to the conductor, making analytic techniques less effective. Their work pushes the boundary of what is analytically tractable in this regime.
4. Decomposition of Character Sums: The detailed decomposition of the character sum $C_{Mc}(m, m')$ using reciprocity and Lemma 5 provides a clear pathway for separating variables in mixed-level settings ($M_1 M_2$).

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4. Main Result

Theorem 1:
Let $M_1$ and $M_2$ be distinct primes with $M_1 < M_2$. Let $\pi$ be a fixed $SL(3, \mathbb{Z})$ Hecke cusp form. Then, for $M_1^\epsilon \leq M_2 \leq M_1^{1+\epsilon}$, the second moment satisfies:
$$\sum_{\substack{\psi \pmod{M_1} \\ \psi(-1)=(-1)^k}} \sum_{f \in H^*_k(M_1 M_2, \psi)} |L(1/2, \pi \times f)|^2 \ll_{\pi, \epsilon} M_1^{1+\epsilon}$$

Interpretation:

• The bound is Lindelöf-consistent. The size of the family is roughly $M_1^2 M_2$. The Lindelöf hypothesis predicts the second moment should be $\approx M_1^2 M_2$.
• The proven bound is $M_1^{1+\epsilon}$. Since $M_2 \approx M_1$ (in the optimal range $M_2 \asymp M_1$), the bound is essentially $M_1^{1+\epsilon}$, which is significantly smaller than the trivial bound and consistent with the conjectured behavior for the family size $|F| \approx M_1^2 M_2 \approx M_1^3$.
• Specifically, the result yields a bound of $M_1 M_2^\epsilon$ (since $M_2 \leq M_1^{1+\epsilon}$), which is a strong subconvexity-type result for the second moment.

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5. Significance

• Advancement in Higher Rank Moments: This paper represents a significant step forward in the study of moments for $GL(3) \times GL(2)$ $L$-functions. Prior to this, such results were largely limited to the spectral aspect (varying the Laplacian eigenvalue) or specific level aspects.
• Non-Vanishing and Subconvexity: Bounds on moments are the primary tool for proving non-vanishing results (showing that $L(1/2, \pi \times f) \neq 0$ for a positive proportion of the family) and establishing subconvexity bounds (bounds better than the convexity bound). This result provides the necessary analytic foundation for these applications in the level aspect.
• Methodological Innovation: The successful integration of $GL(3)$ Voronoi summation with a hybrid large sieve in a level-aspect problem demonstrates a robust framework that could be adapted for other higher-rank automorphic forms or different moment problems.

In summary, the paper resolves a difficult analytic number theory problem by combining deep structural properties of automorphic forms with refined sieve techniques, establishing a sharp upper bound for the second moment of a critical family of $L$-functions.