Exponential sums and Newton polyhedra
Alan Adolphson, Steven Sperber · Bulletin of the American Mathematical Society · 1987
Let p be a prime number and let k denote the field of q = p a elements.Fix a nontrivial additive character V: k -• Q(c p ) x .Given a variety V of dimension n and a regular function ƒ on V, with both V and ƒ defined over &, one can define an exponential sumwhere V(k) denotes the fc-rational points of V.It is a classical problem to find conditions on V and ƒ that will imply a good estimate for |5(V, /)|.By "good estimate" we mean an inequality of the formwhere C is a constant depending on V and ƒ but not on q.Deligne's fundamental theorem [3] reduces the problem of estimating the archimedean size of exponential sums to the problem of computing certain associated /-adic cohomology groups.Let A n denote affine n-space over k and let (G m ) n denote the product of n copies of the multiplicative group G m over k.The purpose of this note is to report on some general criteria, when V = (G m ) n or A n , that allow us to calculate this cohomology and hence obtain sharp archimedean estimates for the corresponding exponential sums.These same criteria allow us to obtain apparently sharp p-adic estimates for the exponential sums as well, although space limitations prevent us from describing them here.Connections between the p-adic theory and Newton polyhedra already appear in [7 and 8].A novel feature of our work is the use of Dwork cohomology [4, 5] to compute /-adic cohomology.The results of this note have not so far been obtainable by purely /-adic methods.Complete proofs and references will appear elsewhere.We are indebted to B. Dwork and N.Katz for many helpful discussions.