\(L\)-functions of symmetric products of the Kloosterman sheaf over \(Z\)
The authors construct explicit virtual schemes \(X\), \(Y\) of finite type over \(\operatorname{Spec}{\mathbb Z}\) such that, for each prime \(p\), the \(p\)-Euler factor of their zeta functions (that is, the zeta function of their reduction modulo \(p\)) is the \(L\)-function of the \(k\)-th tensor power (respectively the \(k\)-th symmetric power) of the Kloosterman sheaf \(\text{Sym}^k{\text{Kl}}_{n+1}\), which is known to be a polynomial [the authors, J. Reine Angew. Math. 589, 79--103 (2005; Zbl 1165.11331)]. This seems surprising at first, since the Kloosterman sheaf is an object which is intrinsically defined in characteristic \(p\) by using an additive character \(\psi\) of the finite field \({\mathbb F}_p\), and can not be lifted to characteristic zero. However, it was shown in [loc. cit.] by the same authors that the global \(L\)-function of its symmetric powers (resp its tensor powers) is a polynomial with \textit{integer} coefficients, and it does not depend on the character chosen. Here they construct a geometric object in characteristic zero which gives rise to these \(L\)-functions for each prime \(p\). This fact adds evidence to the conjecture (known to be true in some cases for \(n=1\)) that the infinite product \[ \prod_p L_p({\mathbb G}_{m,{\mathbb F}_p},\text{Sym}^k{\text{Kl}}_{n+1},s) \] is automorphic and thus extends to a meromorphic function in \(s\in{\mathbb C}\).
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