Exponential sums the geometry of which is very beautiful: \(p\)-adic estimates (Q1175073)

The author studies the \(L\)-function \(L(\bar f,V_{\bar{x}}T)\) associated with twisted exponential sums of the form \(\sum[\prod^ n_{i=1}(\chi_ i\circ N)(t_ i)]\cdot[(\theta\circ Tr)(\bar f(t))]\), where the sum runs over all \(K_ m\)-valued points \((t_ 1,\ldots,t_ n)\) of the affine hypersurface \(V_{\bar x}\) over the finite ground field \(K\) defined by an equation \(t_ 1^{g_ 1}\cdot\ldots\cdot t_ n^{g_ n}=\bar x\;(\bar x\in K^ \times)\); \(K_ m\) is the extension of degree \(m\) of \(K\), \(N\) and \(Tr\) are the norm and trace of \(K_ m| K\), \(\chi_ i\) and \(\theta\) are multiplicative (resp. additive) characters of \(K\), and \(\bar f\) is a polynomial of the form \(c_ 1t_ 1^{k_ 1}+\cdots+c_ nt_ n^{k_ n}\) with coefficients \(c_ i\in K\). Then \(L\) or \(L^{-1}\) (depending on the parity of \(n\)) is a polynomial, and the author computes effectively a lower bound for the Newton polygon of \(L\). Furthermore he shows that this lower bound coincides with the actual Newton polygon if \(p=\text{char}(K)\) lies in certain congruence classes (depending on \(V_{\bar x}\) and \(\bar f\)). This leads to \(p\)-adic estimates for the above twisted exponential sums. The present work generalizes work of Dwork, Sperber, Adolphson-Sperber, and Katz on Kloosterman sums and exponential sums. The main technical tools are a new trace formula and the construction of certain Dwork cohomology space on which a Frobenius acts in such a way that its eigenvalues are the reciprocal zeros of \(L\) (resp. \(L^{-1})\).