Moments and Large Geometric Monodromy for Linearly Constrained Kloosterman Families

Abstract

Let $q=p^a$ with $p$ prime, $\lambda\in\mathbb F_p^\times$, and $\psi_q=\psi\circ\mathrm{Tr}_{\mathbb F_q/\mathbb F_p}$ for a nontrivial additive character $\psi$ of $\mathbb F_p$. For $n\ge2$ we study $\Sigma_n(h)=\sum_{u\in(\mathbb F_q^\times)^n,\sum_i h_i u_i=-\lambda}\psi_q(\sum_i u_i^{-1})$. On $\mathbb G_m^n$ we construct a lisse sheaf $\mathcal W_n$, pure of weight $n+1$ and rank $r(n)=2^n-\binom n{\lfloor n/2\rfloor}$, whose Frobenius traces equal $(-1)^{n-1}q\Sigma_n(h)$ minus an explicit polynomial in $q$, for $h\in(\mathbb F_q^\times)^n$. For odd $p$, its geometric monodromy has identity component $\mathrm{SL}_{r(n)}$ and cyclic component group of order divisible by $p$. The full group is $\{g:\det(g)^p=1\}$ for $n=2$, and for $n\ge3$ using boundary unipotence from the companion preprint arXiv:2610.03081. For $p=2$, the sheaf is self-dual and its geometric monodromy is $\mathrm{SO}_{r(n)}$ for odd $n$ and $\mathrm{Sp}_{r(n)}$ for even $n$. After normalization to weight zero, the arithmetic and geometric groups coincide, using the same companion input for odd $p$ and $n\ge3$. As $q\to\infty$ with $p$ fixed, we obtain Sato-Tate laws for centered, normalized sums with error $O(q^{-1/2})$ for polynomial tests and formulas for all power moments. For $p=2$, limiting moments are Gaussian through order $r(n)-1$, with an excess of one above the orthogonal value at order $r(n)$ for odd $n$; if $n+1$ is a power of two, every Frobenius on $\mathcal W_n$ has eigenvalue $q^{(n+1)/2}$. The proof uses $\sum_h|\Sigma_n(h)|^4=c_pq^{3n-2}+O(q^{3n-2-\delta(n)})$ for $n\ge3$, where $c_p=2$ for odd $p$, $c_2=3$ and $\delta(n)=\min(1,(n-2)/2)$, plane sections of the Cayley cubic, exact Frobenius-Schur identities, Laumon's local Fourier transforms and Artin-Schreier theory. For $n=2$ we identify twisted classical Kloosterman sums and compute the fourth moment exactly.