The weight two and opposite sign cases for the Fourier relative trace formulas

1. Problem Statement and Motivation

The paper addresses a significant gap in the literature regarding the Petersson and Bruggeman-Kuznetsov (BK) trace formulas. While these formulas are fundamental tools in analytic number theory (used for moments, subconvexity, non-vanishing of $L$-functions, and prime distribution), existing adelic proofs via the pre-trace formula (developed by Jacquet, Zagier, Knightly, and Li) have two primary limitations:

1. Sign Restriction: The standard BK formula derived via the pre-trace approach typically handles cases where the Fourier coefficients $m_1$ and $m_2$ have the same sign. The case where $m_1 m_2 < 0$ (opposite signs) was previously lacking a rigorous adelic derivation in this specific framework.
2. Weight Restriction: The holomorphic Petersson formula is well-understood for weights $\kappa > 2$. However, the case $\kappa = 2$ presents analytical difficulties because the matrix coefficient associated with the discrete series $D(2)$ is not in $L^1(G(\mathbb{R}))$, causing the geometric kernel to fail absolute and uniform convergence on compact sets.

The author aims to provide a unified adelic relative trace formula proof for:

• The BK formula in the opposite sign case ($m_1 m_2 < 0$).
• The Petersson formula in the weight two case ($\kappa = 2$).

2. Methodology

The paper employs the Adelic Relative Trace Formula approach, utilizing the pre-trace formula as the starting point. The methodology involves comparing the spectral expansion of a kernel operator with its geometric expansion.

A. The Opposite Sign Case ($m_1 m_2 < 0$)

• Spectral Side: The author analyzes the contribution of cuspidal and continuous spectra. A key feature is the appearance of the parity $p(\pi)$ of the automorphic representation, which arises due to the sign change in the Fourier coefficients.
• Geometric Side: The standard Zagier transform (used for same-sign cases) is insufficient. The author introduces a Modified Zagier Transform, denoted $M(t)$, defined via a kernel $k(z+t, 1/z)$ instead of $k(z+t, -1/z)$.
• Analytic Tool: A crucial step is proving that the Fourier transform of this modified transform, $\hat{M}(a)$, relates to the test function $h_\infty$ via a specific integral involving the $K$-Bessel function and $\sinh(\pi t)$. This is achieved through a contour shift argument to handle convergence issues, establishing the identity:
  $$\hat{M}(a) = \frac{1}{2\pi a} \int_{-\infty}^{\infty} K_{2it}(4\pi a) \sinh(\pi t) t h_\infty(t) dt$$
• Test Functions: The results are derived for a broad class of archimedean test functions $h_\infty$ satisfying specific holomorphic and decay conditions (Paley-Wiener type and extensions).

B. The Weight Two Case ($\kappa = 2$)

• The Obstacle: For $\kappa = 2$, the matrix coefficient of the discrete series $D(2)$ is square-integrable but not absolutely integrable ($L^1$). Consequently, the geometric kernel does not converge absolutely, preventing a direct application of the pre-trace formula.
• The Solution: The author employs a limiting argument using a smooth truncation.
  1. A smooth, bi-$\omega$-isotypic truncation function $\varrho_T$ is introduced, supported on $G(\mathbb{R})^+$, which cuts off the kernel at hyperbolic distance $T$.
  2. The trace formula is applied to the truncated function $f_A^T = f_\infty^T \otimes f$.
  3. The limit $T \to \infty$ is taken. The author rigorously proves that:
     • The contribution from the principal series (continuous spectrum) and the residual spectrum vanishes in the limit.
     • The contribution from the discrete series $D(2)$ converges to the desired spectral sum.
     • The geometric side converges to the diagonal term (involving the delta function) and the off-diagonal terms (involving the Bessel function $J_1$).

3. Key Contributions and Results

A. Refined Bruggeman-Kuznetsov Formula (Opposite Signs)

The paper establishes Theorem 1.2.2, a refined BK formula for $m_1 m_2 < 0$.

• Formula: It equates a spectral sum over cuspidal representations (weighted by parity $p(\pi)$ and Hecke eigenvalues $\lambda_\pi$) and an integral over Eisenstein series to a geometric sum over generalized Kloosterman sums $H(m_1, m_2, c)$.
• Key Feature: The geometric side involves the transform $H_\infty^-$, which is the specific integral transform derived from the modified Zagier transform.
• Assumptions: The formula holds under "Geometric" and "Spectral" assumptions on the non-archimedean test function $f$, allowing for the isolation of newforms and control over the support.

B. Generalized Petersson Formula (Weight 2)

The paper establishes Theorem 1.2.3, the Petersson formula for weight $\kappa = 2$.

• Formula:
  $$\sum_{\pi \in \mathcal{F}_2(f)} w(\pi, f) \lambda_\pi(m_1) \lambda_\pi(m_2) = \frac{1}{4\pi} \delta_{m_1=m_2} \delta - \frac{1}{2} \sum_{c} \frac{H(m_1, m_2, c)}{c} J_1\left(\frac{4\pi\sqrt{m_1 m_2}}{c}\right)$$
• Significance: This resolves the convergence issue for $\kappa=2$ by showing that the "bad" behavior of the kernel is tamed by the limiting argument, recovering the standard diagonal term and the Bessel function term $J_1$.

C. Parity Equidistribution

As a corollary to the opposite sign BK formula, the author proves Corollary 1.3.1, demonstrating the equidistribution of parities for automorphic representations unramified at infinity.

• By choosing specific test functions $h_\infty$ (Gaussian or hyperbolic secant type), the author shows that the sum of weights over representations with parity $\epsilon \in \{0, 1\}$ is asymptotically half the total mass, with a controlled error term. This confirms that even and odd representations are roughly equally distributed in these families.

4. Significance and Impact

• Completeness of the Trace Formula Program: This work completes the adelic derivation of the BK and Petersson formulas for the previously missing cases (opposite signs and weight 2), aligning the modern relative trace formula approach with classical results derived via Poincaré series.
• Applications to Elliptic Curves: The weight $\kappa = 2$ case is of particular importance because it corresponds to modular forms associated with elliptic curves over $\mathbb{Q}$. Having a rigorous trace formula for this weight allows for more precise analytic investigations into the $L$-functions of these curves (e.g., subconvexity bounds, non-vanishing results).
• Flexibility: The use of the adelic framework allows for generalizations to arbitrary number fields and provides a flexible mechanism for selecting families of local representations (via the test function $f$), which is harder to achieve with classical Poincaré series methods.
• Technical Innovation: The introduction of the Modified Zagier Transform and the rigorous limiting argument for the weight 2 case provide new technical tools for handling non-$L^1$ kernels and sign-reversed Fourier coefficients in the context of relative trace formulas.

In summary, Di Scipio's paper provides a rigorous, adelic proof for two critical edge cases in the theory of trace formulas, bridging the gap between classical analytic number theory and modern automorphic representation theory, with direct implications for the study of elliptic curves and $L$-functions.