The shifted convolution problem for Fourier coefficients of Siegel modular forms of degree $2$

TL;DR

The paper resolves the shifted convolution problem for Fourier coefficients of two holomorphic Siegel cusp forms on $ ext{Sp}_4(\mathbb{Z})$ by introducing a novel automorphic reinterpretation of the Duke–Friedlander–Iwaniec delta method. The authors construct a parameterized Poincaré series $P_Q(\cdot,\phi)$ and bound the inner product $\langle F P_Q(\cdot,\phi), F\rangle$ via a reduction to estimating Fourier coefficients of Siegel Poincaré series, leveraging Kitaoka’s bounds on symplectic Kloosterman sums and a careful counting argument. The Fourier expansion of $P_Q$ is decomposed into rank-2, rank-1, and rank-0 contributions, each bounded with precise arithmetic-analytic tools, culminating in the main bound $|\langle F P_Q(\cdot,\phi), F\rangle| \ll N^{5/2+\varepsilon}$ for $Q\neq 0$. This yields a power-saving estimate for shifted convolution sums in the high-rank setting and advances the study of moments and subconvexity for higher-degree $L$-functions via an automorphic delta-method framework.

Abstract

We provide a power-saving bound for certain smoothed shifted convolution sums for Fourier coefficients of Siegel cusp forms. This result is the first nontrivial estimate for a shifted convolution sum with two cusp forms on a group of higher rank than $\GL_2$. Our approach is based on a novel automorphic reinterpretation of the delta method of Duke, Friedlander, and Iwaniec. The method reduces the problem to the estimation of Fourier coefficients of Siegel Poincare series, which is ultimately based on the Weil bound.

Figures (2)

• Figure 1: Polar lines of $E_0^*(g, \nu_1, \nu_2)$
• Figure 2: Polar lines of $P(\nu_1, \nu_2) \cdot E_0^*(g, \nu_1, \nu_2)$

Theorems & Definitions (51)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2: Kitaoka's bound
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof