Effective quantitative orthogonality of Fourier coefficients of Siegel-Poincare series and Satake-parameter equidistribution for genus $g\ge 3$ Siegel cusp forms in the weight aspect

A short 2011 note of Kowalski-Saha-Tsimerman proved, by a soft dominated-convergence argument, the asymptotic orthogonality of Fourier coefficients of Poincare series for classical modular forms (in the weight and level aspects) and for Siegel modular forms (Theorem 4: the coefficient $p_{s,k}(t)$ of the genus-$g$ Poincare series tends, as the even weight $k\to\infty$, to a Kronecker delta on $\mathrm{GL}(g,\mathbb Z)$-equivalence classes of the matrix index, times $|\mathrm{Aut}(s)|/2$). The note itself flagged the gaps that its qualitative method leaves open: no error terms or effective constants; an integration-cycle height $y_0$ known only to exist; and Siegel analogues of the GL(2) strong-approximation corollary, deferred to the authors' companion work. Later literature closed most of this agenda: the companion paper (Compositio 2012) settled the genus-2 weight aspect with a quantitative orthogonality formula of power-saving strength (Prop 3.3) and quantitative local equidistribution of spinor Satake parameters; the level aspect was settled for degree 2 (Dickson) and, via trace formulas, for all degrees (Knightly-Li; Kim-Wakatsuki-Yamauchi); Anamby-Das (2023) resolved the optimal-height question by proving that every $y_0>1$ is admissible for every genus; and Das (2024) proved, for all degrees, an effective asymptotic for the Poincare coefficients whose error term decays in $k$ but grows exponentially in the matrix index, so that it is uniform only up to index scales linear in $k$. The same paper confirms, for all degrees, the $L^\infty$-size conjectures for the spaces of cusp forms, by a route that avoids the Petersson trace formula. What remains open, and is documented as unavailable in the most recent literature (Das-Krishna, IMRN 2024, and still in Das 2024), is the degree $g\ge 3$ weight-aspect core: a quantitative orthogonality formula for Siegel-Poincare Fourier coefficients uniform in the matrix index at the polynomial-in-$k$ scales that the spectral applications require -- the higher-rank analogue of the classical Kloosterman-sum and Bessel-function terms of the Petersson formula -- together with the Fourier-coefficient-to-Satake-parameter bridge needed for quantitative local spectral equidistribution of degree $\ge 3$ families. The problem is to establish these.

Background

A holomorphic cusp form of even weight $k$ for a congruence subgroup $\Gamma\subset\mathrm{SL}(2,\mathbb Z)$ is a holomorphic function on the upper half-plane $\mathbb H=\{x+iy:y>0\}$ that vanishes at the cusps and satisfies $f(\gamma z)=(cz+d)^k f(z)$ for $\gamma=\bigl(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\bigr)$ in $\Gamma$. Such forms carry Fourier expansions $f(z)=\sum_{n\ge 1}a_f(n)e(nz)$ with $e(z)=e^{2\pi i z}$, and are simultaneously eigenfunctions of Hecke operators whose eigenvalues $\lambda_f(n)$ encode the $L$-function of $f$. The $m$-th Poincare series $P_{m,k}(z)=\sum_{\gamma\in\Gamma_\infty\backslash\Gamma}(cz+d)^{-k}\,e(m\,\gamma\cdot z),$ where $\Gamma_\infty$ is the stabilizer of the cusp at infinity, is an explicit cusp form whose Fourier coefficients $p_{m,k}(n)$ are, by adjointness of Hecke operators with respect to the Petersson inner product (the standard $L^2$ inner product on cusp forms), the spectrally weighted average of $\lambda_f(m)$ over a basis of cusp forms. The Petersson formula evaluates $p_{m,k}(n)$ as a main term (the Kronecker delta $\delta(m,n)$) plus off-diagonal terms built from Kloosterman sums $S(m,n;c)=\sum_{r\bmod c}^{*}e((mr+n\bar r)/c)$ (the star restricting $r$ to be invertible mod $c$) twisted by Bessel functions $J_{k-1}$. Quantitative control of the off-diagonal terms -- decay in the weight $k$ or the level $q$, uniform in $m$ and $n$ -- is a workhorse of the analytic theory of families of automorphic forms and $L$-functions: it underlies Serre's equidistribution of Hecke eigenvalues in the weight aspect and Katz-Sarnak-type statements on the symmetry type of a family (the statistics of the low-lying zeros of its $L$-functions).

Kowalski-Saha-Tsimerman (Mathematika 57, 2011) observed that the qualitative core of this circle -- $\lim_{k\to\infty}p_{m,k}(n)=\delta(m,n)$, and the level-aspect analogue as the level $q\to\infty$ -- admits a short soft proof, and extended it to Siegel modular forms. A Siegel cusp form of genus $g\ge 1$ and even weight $k$ for $\Gamma_g=\mathrm{Sp}(2g,\mathbb Z)$ is a holomorphic function on the Siegel upper half-space $\mathbb H_g=\{z=x+iy:{}^tz=z,\ y\ \text{positive definite}\}$ (symmetric complex $g\times g$ matrices with positive-definite imaginary part), transforming by $\det(cz+d)^k$ and vanishing at the rational boundary; its Fourier expansion is indexed by positive-definite symmetric half-integral matrices $s$ (the matrix indices): $F(z)=\sum_{s>0}a_F(s)e(\mathrm{Tr}(sz))$. For such $s$ the genus-$g$ Poincare series $P_{s,k}$ has Fourier coefficients $p_{s,k}(t)$, and Theorem 4 of the note states that, as $k\to\infty$ over even $k$, $p_{s,k}(t)$ tends to $\delta'(s,t)\,|\mathrm{Aut}(s)|/2$, where $\delta'$ is the Kronecker delta for $\mathrm{GL}(g,\mathbb Z)$-equivalence classes of indices and $\mathrm{Aut}(s)$ is the group of integral matrices $u$ with ${}^tusu=s$ (generically of order 2 for $g\ge 3$, by Witt's cancellation theorem for quadratic forms, but possibly larger for special indices). The soft proof integrates over the horizontal cycle $U_g(y_0)=\{x+i\,y_0\mathrm{Id}:|x_{ij}|\le 1/2\}$, whose height $y_0$ must satisfy $|\det(cz+d)|\ge 1$ for all symplectic $\gamma$ with $c\ne 0$; the note's Lemma 5 produces some such $y_0>1$ from Siegel's fundamental-domain theory (Minkowski reduction of the imaginary part; a finite determining set of pairs $(c,d)$, explicit only for $g\le 2$ via Gottschling).

The note is explicit that these soft statements are a prototype, not a tool: they carry no error terms and no effective constants, whereas the deeper applications (averages of $L$-functions in the critical strip, equidistribution with error terms) require quantitative orthogonality. At genus 1 the quantitative theory is classical; Remark 8 of the note defers the genus-2 case, with quantitative formulations, to the authors' companion paper. The representation-theoretic target of such relations is the local spectral equidistribution of Satake parameters: the multiset of complex parameters attached to the unramified local components $\pi_p$ of the automorphic representations in the family, which determine all local $L$-factors. From quantitative equidistribution follow analogues of strong approximation for cusp forms (density of tuples of local components in the tempered unramified spectrum, 'tempered' meaning that all Satake parameters have complex modulus one; for $\mathrm{GL}(2)$ this is due to Serre for holomorphic forms and to Sarnak for Maass forms) and the symmetry type of the family's $L$-functions. The link between Fourier coefficients and Satake parameters is elementary at genus 1 but involves the spinor $L$-function (the degree-$2^g$ $L$-function attached to a genus-$g$ cusp form; at genus 1 it is the usual $L$-function of a classical cusp form) and delicate coefficient relations at genus 2, and is structurally harder at genus $g\ge 3$, where the degree-$2^g$ spinor $L$-function's Euler factors mix several Fourier coefficients.

Subsequent literature resolved most of this agenda: the genus-2 weight aspect (Kowalski-Saha-Tsimerman, Compositio 148, 2012: quantitative orthogonality with power-saving error uniform in the index, via an adaptation of Kitaoka's degree-2 coefficient estimates, and quantitative local equidistribution of the spinor Satake parameters with error terms); the level aspect in all degrees (Knightly-Li via an asymptotic Petersson formula for $\mathrm{GSp}(2n)$; Kim-Wakatsuki-Yamauchi via Arthur's invariant trace formula); the optimal height (Anamby-Das: every $y_0>1$ is admissible for every genus, resolving the note's Remark 7); and an effective asymptotic for the Poincare coefficients in all degrees, whose error term, however, grows exponentially in the matrix index (Das, 2024: uniform only up to index scales linear in $k$; the same paper confirms the $L^\infty$-size conjectures for the spaces of cusp forms in all degrees, by a route that avoids the Petersson trace formula). For genus $g\ge 3$ in the weight aspect -- the aspect of the source note -- no quantitative orthogonality formula with polynomial (or otherwise summable) index dependence and no Satake-parameter equidistribution result is known; Das-Krishna (IMRN 2024) document precisely this gap (insufficient knowledge of the matrix-argument Bessel functions and matrix-index Kloosterman sums, and their Lemma 4.1 'is not available in the literature if the uniformity with respect to $k$ is requested'), and Das (2024) still concurs that 'the explicit formula for such Fourier coefficients is not available for higher degrees'.

Problem Statement

For Siegel cusp forms of genus $g\ge 3$ for $\mathrm{Sp}(2g,\mathbb Z)$ (or for specified congruence subgroups), in the weight aspect $k\to\infty$ over even weights, establish the effective quantitative theory whose qualitative prototype is Theorem 4 of Kowalski-Saha-Tsimerman (Mathematika 57 (2011) 31-40). Concretely:

(1) Quantitative orthogonality of Poincare-series Fourier coefficients: for the Fourier coefficients $p_{s,k}(t)$ of the genus-$g$ Poincare series with $s$ not $\mathrm{GL}(g,\mathbb Z)$-equivalent to $t$, prove an explicit asymptotic formula or decaying bound, with an error term that is effective in $k$ and uniform in the matrix indices $s,t$ over a range that grows with $k$ -- specifically, at the polynomial-in-$k$ index scales that arise when the kernel is summed against the Fourier coefficients of the cusp forms of the family, as in the approximate functional equations of the spinor and standard $L$-functions (at degree 2 the companion paper's applications require determinant parameters ranging over powers of $k$); an error term growing exponentially in the index, effective only up to index scales linear in $k$, does not meet this requirement. This is the degree-$g$ analogue of the classical quantitative Petersson formula, whose off-diagonal terms are Kloosterman sums twisted by Bessel functions, and its degree-2 model is Proposition 3.3 of the authors' companion paper (Compositio 148 (2012)), with its power-saving error uniform in the index parameters.

(2) Satake bridge and local equidistribution: using (1) or any other valid route, identify the Hecke/Satake data at finitely many primes through Fourier coefficients of bounded-determinant indices, and prove the quantitative local spectral equidistribution of the Satake parameters of the degree-$g$ family as $k\to\infty$, for a suitable harmonic weighting, with an explicit error term and an identified limiting measure (Sato-Tate measure on the relevant maximal torus); derive the corresponding analogue of the strong-approximation corollary (density of tuples of local components in the tempered unramified spectrum) and, if possible, the symmetry type of the associated family of spinor or standard $L$-functions.

(1) and (2) are two aspects of one objective -- the effective weight-aspect family theory for degree $\ge 3$ Siegel cusp forms -- with (1) supplying the analytic input used by (2) and the degree-$g\ge 3$ case being the analogue of what the Compositio 2012 companion established at degree 2. (The auxiliary optimal-height ingredient of the soft proof -- Remark 7 of the source note -- is already resolved: Anamby-Das proved every $y_0>1$ admissible for every genus, so the infimal admissible height is 1; it is therefore not part of the problem.)

References

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