$L^\infty$-sizes of the spaces Siegel cusp forms of degree $n$ via Poincaré series

1. Problem Statement

The paper addresses the sup-norm problem for Siegel cusp forms of degree $n \geq 1$, weight $k$, and level $N$. Specifically, it investigates the $L^\infty$-size (or "size") of the space of Siegel cusp forms, denoted as $S_k^{(n)}(\Gamma)$, where $\Gamma$ is a congruence subgroup of the symplectic group $Sp_n(\mathbb{Z})$.

The size is measured by the supremum of the Bergman kernel $B_{k,\Gamma}(Z)$ over the Siegel upper half-space $\mathbb{H}_n$:
$$\text{sup}(S_k^{(n)}(\Gamma)) := \sup_{Z \in \mathbb{H}_n} B_{k,\Gamma}(Z) = \sup_{Z \in \mathbb{H}_n} \sum_{F \in \mathcal{B}_k(\Gamma)} \det(Y)^k |F(Z)|^2$$
where $\mathcal{B}_k(\Gamma)$ is an orthonormal basis of $S_k^{(n)}(\Gamma)$ with respect to the Petersson inner product.

The paper aims to prove Conjecture 1.1, which posits that as the weight $k \to \infty$:
$$\text{sup}(S_k^{(n)}(\Gamma)) \asymp_{n, \Gamma} k^{\frac{3n(n+1)}{4}}$$
This is the "average" sup-norm problem. The paper also addresses the "individual" sup-norm problem for Hecke eigenforms (Conjecture 1.3), suggesting $\|F\|_\infty \ll k^{\frac{n(n+1)}{8}}$.

2. Methodology

The author employs a method based on the Fourier expansion of the Bergman kernel and the properties of Poincaré series, avoiding the heat-kernel methods used in recent literature (e.g., [KM23]) and the complex geometric analysis of the Bergman kernel used in [DK24].

Key Technical Steps:

1. Geometric Expression of the Bergman Kernel:
   The Bergman kernel is expressed using the Fourier expansion of Poincaré series $P_{T, \Gamma}(Z)$:
   $$B_{k,\Gamma}(Z) \propto \sum_{T \in \Lambda_\Gamma} \det(T)^{k - \frac{n-1}{2}} P_{T, \Gamma}(Z) e(-TZ)$$
   where $\Lambda_\Gamma$ is the lattice of half-integral positive semi-definite matrices associated with $\Gamma$.

2. Bounding Poincaré Series:
   The core of the proof involves showing that for $Z$ in the fundamental domain $\mathcal{F}_1^{(n)}$ (level 1), the Poincaré series are uniformly bounded:
   $$P_{T, \Gamma}(Z) \ll_n 1$$
   This is achieved by:

   • Decomposing the sum over $\Gamma$ into cosets of the translation subgroup $\Gamma_\infty$.
   • Utilizing the Lipschitz formula to relate sums over lattices to theta series.
   • Proving that the sub-series $H_\Gamma(T, Y) = \sum_{U \in U_\Gamma} \exp(-2\pi \text{tr}(TY[U]))$ decays rapidly enough to counteract the growth of the automorphy factor $J(g, Z)^{-k}$.

3. Handling Conjugates and Levels:
   For a general congruence subgroup $\Gamma$, the fundamental domain is a union of translates of the level-1 domain. The author uses a conjugation argument:
   $$\sup(S_k^{(n)}(\Gamma)) = \max_j \sup(S_k^{(n)}(g_j^{-1}\Gamma g_j))$$
   By bounding the Poincaré series for all conjugates $G = g_j^{-1}\Gamma g_j$, the author derives uniform bounds.

4. Lower Bounds via Asymptotics:
   To establish the lower bound (proving the size is at least $k^{\frac{3n(n+1)}{4}}$), the author derives a quantitative asymptotic formula for the Fourier coefficients of Siegel Poincaré series (Theorem 5.1). This formula shows that for large $k$, the diagonal Fourier coefficient $p_T(T)$ approaches a non-zero constant related to the automorphism group of $T$.

3. Key Contributions and Results

A. Main Theorems

• Theorem 1.5 (Weight Aspect): For any congruence subgroup $\Gamma$ of level $N$ and weight $k \geq 2n+2$:
  $$c_\Gamma k^{\frac{3n(n+1)}{4}} \ll_n \text{sup}(S_k^{(n)}(\Gamma)) \ll_{n, \epsilon} k^{\frac{3n(n+1)}{4}} N^{\frac{n^2}{2} + \epsilon}$$
  This confirms the conjectured growth rate in the weight aspect for all familiar congruence subgroups.

• Theorem 1.6 (Hybrid/Level Aspect for Principal Subgroups): For the principal congruence subgroup $\Gamma^{(n)}(N)$, the bound is uniform in the level aspect (for fixed $k$):
  $$\text{sup}(S_k^{(n)}(\Gamma^{(n)}(N))) \asymp_n k^{\frac{3n(n+1)}{4}}$$
  This removes the dependence on the level $N$ in the upper bound, a significant improvement over Theorem 1.5 for principal subgroups.

• Theorem 1.7 (Hybrid Aspect with Large Weight): If the weight is sufficiently large relative to the level ($k > \max\{(n+1)^2, N^{3+\epsilon}\}$), the uniform bound holds for any congruence subgroup:
  $$\text{sup}(S_k^{(n)}(\Gamma)) \asymp_n k^{\frac{3n(n+1)}{4}}$$

B. Applications

1. Non-vanishing of Poincaré Series:
   The author proves that Siegel Poincaré series of exponential type do not vanish identically for sufficiently large weights. Specifically, for a set of pairwise inequivalent matrices $\{T_1, \dots, T_h\}$, the corresponding Poincaré series are linearly independent if $k$ is large enough (Corollary 5.8). This generalizes previous results from degree $n=2$ to arbitrary $n$.

2. Small Weights:
   Using a norm interpolation technique (Hölder's inequality) and the "weight-embedding" method, the paper extends the sup-norm bounds to small weights ($k \geq n/2$) under the assumption that the average size bound holds for large weights (Theorem 9.2).

3. Elliptic Modular Forms ($n=1$):
   The appendix provides a new proof for the size of spaces of elliptic cusp forms, including a large sieve inequality for arbitrary congruence subgroups at weight $k=2$, yielding a bound of $O(\log N)$.

4. Significance

• Resolution of Conjectures: The paper provides the first complete proof of the conjectured asymptotic size $k^{\frac{3n(n+1)}{4}}$ for Siegel cusp forms in the weight aspect for general degree $n$ and arbitrary congruence subgroups.
• Simplicity and Generality: Unlike previous approaches that relied on complicated heat-kernel analysis or specific geometric properties of the fundamental domain, this method relies on the Fourier expansion of Poincaré series. This makes the arguments more accessible and applicable to vector-valued forms, half-integral weights, and other automorphic forms on tube domains.
• Level Independence: The result for principal congruence subgroups (Theorem 1.6) demonstrates that the "size" of the space is intrinsic to the weight and degree, independent of the level, once the level is fixed.
• New Tools: The derivation of the quantitative asymptotic formula for Fourier coefficients of Siegel Poincaré series (Theorem 5.1) is a significant independent result, enabling the proof of non-vanishing and linear independence results without explicit formulas for the coefficients.

In summary, Soumya Das establishes a robust framework for analyzing the $L^\infty$-norms of Siegel cusp forms, confirming long-standing conjectures and providing a versatile toolkit for future research in the analytic theory of automorphic forms.