L-functions of Symmetric Products of the Kloosterman Sheaf over Z

The classical $n$-variable Kloosterman sums over the finite field ${\bf F}_p$ give rise to a lisse $\bar {\bf Q}_l$-sheaf ${\rm Kl}_{n+1}$ on ${\bf G}_{m, {\bf F}_p}={\bf P}^1_{{\bf F}_p}-\{0,\infty\}$, which we call the Kloosterman sheaf. Let $L_p({\bf G}_{m,{\bf F}_p}, {\rm Sym}^k{\rm Kl}_{n+1}, s)$ be the $L$-function of the $k$-fold symmetric product of ${\rm Kl}_{n+1}$. We construct an explicit virtual scheme $X$ of finite type over ${\rm Spec} {\bf Z}$ such that the $p$-Euler factor of the zeta function of $X$ coincides with $L_p({\bf G}_{m,{\bf F}_p}, {\rm Sym}^k{\rm Kl}_{n+1}, s)$. We also prove similar results for $\otimes^k {\rm Kl}_{n+1}$ and $\bigwedge^k {\rm Kl}_{n+1}$.Comment: 16 page

On the p-adic local invariant cycle theorem

For a proper, flat, generically smooth scheme X over a complete discrete valuation ring with finite residue field of characteristic p, we construct a specialization morphism from the rigid cohomology of the geometric special fibre to D-cris of the p-adic etale cohomology of the geometric generic fibre, and we make a conjecture ("p-adic local invariant cycle theorem") that describes the behavior of this map for regular X, analogous to the situation in l-adic etale cohomology for l not equal p. Our main result is that, if X has semistable reduction, this specialization map induces an isomorphism on the slope [0, 1)-part