The cubic moment of $L$-functions for specified local component families

1. Problem Statement

The paper addresses the subconvexity problem for central values of $L$-functions associated with automorphic representations of $PGL_2(\mathbb{Q})$. Specifically, it aims to establish Weyl-strength subconvex bounds (saving a factor of $Q^{1/12-\epsilon}$ over the convexity bound) for families of forms where the local components at finitely many primes are fixed.

• Context: Previous work (e.g., Conrey-Iwaniec, Petrow-Young) successfully handled families where the local components were in the principal series or were ramified twists of the Steinberg representation.
• The Gap: These methods failed to cover the supercuspidal case. Supercuspidal representations correspond to "twist-minimal" forms, which constitute a significant portion of the space of automorphic forms (specifically, those that cannot be obtained by twisting lower-level forms).
• Goal: To prove Lindelöf-on-average upper bounds for the cubic moment $M_3(\mathcal{F}) = \sum_{f \in \mathcal{F}} |L(1/2, f)|^3$ for families $\mathcal{F}$ defined by specifying supercuspidal local components. This would imply subconvexity for these previously inaccessible families.

2. Methodology

The authors employ a sophisticated analytic number theory approach, combining the approximate functional equation, the Petersson/Bruggeman-Kuznetsov (PBK) trace formula, and algebraic geometry.

A. New PBK Formulas for Supercuspidal Families

The core technical innovation is the utilization of recently developed PBK formulas (from the authors' previous work [Hu24, HPY26]) tailored to local component families.

• Unlike classical trace formulas that sum over all forms of a given level, these formulas isolate forms with specific local behavior at primes $p$ dividing the conductor.
• The formula relates the spectral sum to a generalized Kloosterman sum $H(m, n; c)$.
• For supercuspidal representations $\sigma_p$, these sums are expressed via the Local Langlands Correspondence (LLC) in terms of a quadratic extension $L/\mathbb{Q}_p$ and a character $\xi$ of $L^\times$.

B. The Motohashi-Type Formula

The proof follows the "Motohashi" strategy:

1. Expansion: Expand $L(1/2, f)^3$ using an approximate functional equation.
2. Trace Formula: Apply the PBK formula to swap the sum over automorphic forms for a sum over Kloosterman sums.
3. Poisson Summation: Apply Poisson summation in three variables to the resulting sum.
4. Dual Moment: This transforms the problem into estimating a dual moment of Dirichlet $L$-functions:
   $$\sum_{\psi} |L(1/2, \psi)|^4 g(\xi, \psi)$$
   where $g(\xi, \psi)$ is a complex character sum derived from the supercuspidal Kloosterman sums.

C. Analysis of Character Sums (The "Hard" Part)

The critical difficulty lies in bounding the character sum $g(\xi, \psi)$ (denoted as $\tilde{H}(\psi)$ in the paper).

• Generic Case: For generic characters, the sum exhibits square-root cancellation.
• Singular Case: The sum can be large only for "singular" characters $\psi$ satisfying specific congruence conditions related to the extension $L/\mathbb{Q}_p$.
• Algebraic Geometry: To bound the singular cases, the authors interpret the character sums as $\ell$-adic trace functions of sheaves on $\mathbb{P}^1$.
  • They utilize the Deligne-Laumon geometric Fourier transform.
  • They construct specific sheaves (Kloosterman sheaves and Kummer sheaves) and prove they are geometrically irreducible and have bounded conductors.
  • By applying the Riemann Hypothesis for varieties over finite fields (Deligne's theorem) and orthogonality relations for trace functions, they establish the necessary square-root cancellation bounds.

D. Handling the Depth Aspect

The paper treats two main scenarios:

1. Square-full Level: $Q$ is a perfect square.
2. Depth Aspect: $Q = p^\alpha$ for a fixed prime $p$ and large $\alpha$.
   The analysis distinguishes between unramified and ramified quadratic extensions $L/\mathbb{Q}_p$, as the structure of the character sums differs significantly (e.g., the behavior of the Postnikov formula and the resulting congruences).

3. Key Contributions

1. First Treatment of Supercuspidal Cubic Moments: This is the first paper to successfully derive Weyl-strength subconvex bounds for cubic moments where the local components are supercuspidal.
2. New PBK Formulas: The authors provide and utilize explicit PBK formulas for families defined by supercuspidal local components, which are essential for the trace formula step.
3. Geometric Fourier Transform Application: The paper demonstrates a powerful application of the Deligne-Laumon geometric Fourier transform to bound character sums arising from supercuspidal representations, a technique previously less common in this specific analytic context.
4. Resolution of the "Twist-Minimal" Gap: It completes the subconvexity picture for $PGL_2$ forms of square-full level, covering the "twist-minimal" forms that were previously out of reach.

4. Main Results

Theorem 1.1 (Lindelöf-on-Average Bound)

For any local component family $\mathcal{F}$ (defined by specifying supercuspidal local components at a finite set of primes), the cubic moment satisfies:
$$M(\mathcal{F}) \ll_\epsilon |\Pi_\infty|^B Q \prod_{p} p^{\lceil \alpha_p/2 \rceil (1+\epsilon)}$$
If the spectral parameters are bounded away from zero, this implies the Lindelöf-on-average bound $M(\mathcal{F}) \ll_\epsilon |\mathcal{F}|^{1+\epsilon}$.

Corollary 1.2 (Subconvexity for Square-Full Level)

Let $f$ be a cuspidal newform of square-full level $Q = \prod p^{\alpha_p}$ with trivial central character. Then:
$$L(1/2, f) \ll_\epsilon Q_\infty^{1/6} \left( \prod_{p} p^{\frac{1}{3} \lceil \alpha_p/2 \rceil} \right) (Q_\infty Q)^\epsilon$$

• Special Case 1 (Odd Square): If $Q$ is an odd perfect square, this yields the Weyl bound: $L(1/2, f) \ll_\epsilon (Q_\infty Q)^{1/6+\epsilon}$.
• Special Case 2 (Prime Power): If $Q = p^\alpha$, this yields the Weyl bound in the depth aspect: $L(1/2, f) \ll_{\epsilon, p} (Q_\infty p^\alpha)^{1/6+\epsilon}$.
• Level $p^2$: The result implies a Weyl-subconvex bound for all cusp forms of level $p^2$. Previously, such bounds were only known for forms that were twists of level $p$ forms (roughly half the space).

5. Significance

• Completeness: This work effectively completes the program initiated by Conrey and Iwaniec for cubic moments, covering all local component types (Principal, Special, and Supercuspidal) for $PGL_2$.
• Methodological Advancement: It bridges the gap between analytic number theory (trace formulas, moments) and arithmetic geometry (sheaf theory, $\ell$-adic cohomology). The ability to handle supercuspidal representations via geometric Fourier transforms opens new avenues for studying $L$-functions with more complex local structures.
• Applications: The derived subconvex bounds have immediate applications in problems involving the distribution of values of $L$-functions, quantum unique ergodicity, and the non-vanishing of central values. Specifically, it resolves the subconvexity problem for the "hardest" remaining cases of square-full levels.

In summary, the paper represents a major breakthrough in the analytic theory of $L$-functions by overcoming the technical barriers associated with supercuspidal representations through the synthesis of new trace formulas and deep results from algebraic geometry.