Geometrization of summation formulae for quadrics

Technical Summary: Geometrization of Summation Formulae for Quadrics

Problem Statement
The paper addresses the Poisson summation formula for the zero locus $X_\ell$ of a split quadratic form in an even number of variables ($2\ell$) over a number field $E$. While a Poisson summation formula was previously established for $\ell \ge 3$ (Theorem 1.1 in [Hsu21, Get25]), it involved "boundary terms" (residues of degenerate Eisenstein series) that were not explicitly expressed in geometric terms. Specifically, the residues at $s = \pm 1$ and $s = \pm (\ell-1)$ were known to vanish if and only if certain boundary maps applied to the Schwartz function were zero, but the non-zero values of these residues lacked a geometric description.

In the function field case, a complete geometric description of these residues was achieved in [GK24] using the automorphy of minimal representations. However, the number field case presents distinct challenges due to archimedean places. A recent approach by Getz [Get25] introduced a function space $\tilde{S}(X_\ell(\mathbb{A}_E))$ defined via theta lifts and a linear map $I$ to smooth functions on the smooth locus. While [Get25] established a geometric summation formula for functions of the form $I(f)$, it remained unclear whether $I(f)$ actually belongs to the standard Schwartz space $S(X_\ell(\mathbb{A}_E))$ and whether the formula descends to depend solely on the image $I(f)$.

Methodology
The author resolves these ambiguities by establishing a rigorous equivalence between two distinct constructions of Schwartz spaces on quadrics:

1. Braverman-Kazhdan Spaces: The standard construction of $S(X_\ell)$ via minimal representations of the orthogonal group $O_{V_{\ell+1}}$, as developed in [GHL25] and [Hsu21].
2. Theta Lift Spaces: The space $\tilde{S}(X_\ell)$ defined via the coinvariants of the Weil representation of $SL_2$ acting on $S(V_\ell \oplus \mathbb{A}^2)$, as introduced in [Get25].

The methodology proceeds in three main stages:

• Local Comparison: The paper analyzes the local theory over both non-archimedean and archimedean local fields. It explicitly computes the asymptotic behavior of functions near the singularity (the origin) for both constructions.
• Identification of Maps: The author proves that the map $I: \tilde{S}(X_\ell) \to C^\infty(X_\ell^\circ)$ is an isomorphism onto the standard Schwartz space $S(X_\ell)$. Crucially, this isomorphism intertwines the Fourier transforms defined in both theories ($\tilde{F}_{X_\ell}$ and $F_{X_\ell}$) and the boundary maps (residues and asymptotic coefficients).
• Global Descent: Using the local identifications, the author lifts the summation formula from the theta lift setting to the standard Schwartz space setting. This involves carefully tracking normalization constants, particularly the discriminant $D$ of the number field and the epsilon factors $\epsilon(s, \psi)$, to ensure the adelic formula holds.

Key Contributions and Results

1. Equivalence of Schwartz Spaces: The paper proves that for $\ell \ge 2$, the image of the theta lift map $I$ is exactly the standard Schwartz space $S(X_\ell(\mathbb{A}_E))$ (Theorem 7.1, Propositions 5.2, 6.3). This confirms that the theta lift construction provides a valid model for the Schwartz space in the number field setting.
2. Geometrization of Residues: The primary result is Theorem 1.2, a "geometrized" Poisson summation formula. This formula expresses the sum of a Schwartz function over rational points on the quadric in terms of:
   • The sum of its Fourier transform.
   • Explicit boundary terms involving iterated boundary maps $d_{\ell, i}$ and $c_i$.
   • These boundary maps correspond to the asymptotic expansions of the function near the origin (germs).
3. Explicit Residue Formulas: By comparing the new formula (Theorem 1.2) with the previous analytic formula (Theorem 1.1), the author derives explicit geometric expressions for the residues of the degenerate Eisenstein series:
   $$\text{Res}_{s=-(\ell-1)} \text{Eis}(f_s) = \kappa |D|^{\frac{\ell-3}{2}} \zeta(\ell-1) c_\ell(f)$$
   and a similar expression for the residue at $s=-1$ involving the sum of terms related to $d_\ell$.
4. Handling of $\ell=2$: The paper provides a detailed analysis for the case $\ell=2$, where the Eisenstein series may have a pole of order 2. It introduces an additional boundary map $a_2$ to capture the coefficient of the $(s+1)^{-2}$ term, which was not present in the $\ell \ge 3$ case.

Significance
The paper claims that its main contribution is providing a geometric expression for the boundary terms in the Poisson summation formula for quadrics over number fields. By showing that these terms are determined by the germs of functions at the singularity (via the maps $c_\ell$ and $d_\ell$), the work "geometrizes" the analytic residues.

The author notes that this result is akin to regularized Siegel-Weil identities (citing [Ich01, Ike96, KR94]). The significance lies in demonstrating that such identities can be used to geometrize boundary terms for Braverman-Kazhdan spaces. The paper suggests this approach could be extended to other settings, such as rephrasing Poisson summation formulae on the Lagrangian Grassmannian (as in [GL21]) in a geometric fashion, though it does not perform this extension itself.

In summary, the paper bridges the gap between the analytic theory of Eisenstein series and the geometric theory of Schwartz spaces on singular varieties, confirming that the theta lift approach of [Get25] is fully compatible with the Braverman-Kazhdan framework and yields a complete, geometric description of the summation formula's error terms.