Cohomology of Linearly Constrained Kloosterman Families: Newton Polytopes, Weights, and Boundary Monodromy

Abstract

Let $p$ be an odd prime, $\psi$ a nontrivial additive character of $\mathbb F_p$, $\psi_q=\psi\circ\mathrm{Tr}_{\mathbb F_q/\mathbb F_p}$ and $\lambda\in\mathbb F_p^\times$. For $h\in(\mathbb F_q^\times)^n$ let $\Sigma_n(h)$ be the sum of $\psi_q(u_1^{-1}+\cdots+u_n^{-1})$ over $u\in(\mathbb F_q^\times)^n$ with $h_1u_1+\cdots+h_nu_n=-\lambda$, an additively twisted average of products of Kloosterman sums. For every $n\ge2$, the cohomology computing $\Sigma_n(h)$ is concentrated in one degree and has dimension $2^n-1$. It has $r(n)=2^n-\binom n{\lfloor n/2\rfloor}$ eigenvalues of weight $n-1$, and its other eigenvalues are the integers $q^{k-1}$, $1\le k\le\lfloor n/2\rfloor$, with ballot numbers as multiplicities. The part of top weight forms a lisse sheaf $\mathcal W_n$ of rank $r(n)$ on the torus $\mathbb G_m^n$. We determine its local monodromy at the coordinate boundary and its determinant: $\det\mathcal W_n$ is the Artin--Schreier sheaf of the sum $-2^{n-1}\lambda^{-1}(h_1+\cdots+h_n)$ of the critical values of the phase, twisted by $p^{(n+1)r(n)/2}$. The geometric monodromy group of $\mathcal W_n$, and the arithmetic one after normalization to weight $0$, are equal to $\{g\in\mathrm{GL}_{r(n)}:\det(g)^p=1\}$. Consequently the normalized sums become equidistributed like traces of Haar-random elements of $\{g\in\mathrm U(r(n)):\det(g)^p=1\}$. The proofs use Laumon's Fourier transform, middle convolution, the Cayley cubic surface and Larsen's alternative, but no classification of finite groups.