Search

We provide an effective version of Katz’ criterion for finiteness of the monodromy group of a lisse, pure of weight zero, ℓ-adic sheaf on a normal variety over a finite field, depending on the numerical complexity of the sheaf.

Given a Galois étale map of varieties p:Y?Xand an l-adic sheaf or complex on Y, we study two cohomological operations: the tensor direct image and the convolution direct image (in the case of perverse sheaves), which give objects on X, and can be used to improve the estimates on some partial exponential sums and the number of rational points on certain varieties.

AbstractWe give a purity result for exponential sums of the type , where kis a finite field of characteristic p, ψ : k→ is a non-trivial additive character and ƒ ∈ kx1,…, xn is a polynomial whose highest degree homogeneous form splits as a product of factors defining a divisor with normal crossings in n−1.

We give a purity result for two kinds of exponential sums of the type $\sum_{x\in k^n}\psi(f(x))$, where k is a finite field of characteristic p and $\psi:k\to{\mathbb C}^\star$ is a non-trivial additive character. In the first case, $f\in k[x_1,\dots,x_n]$ is a polynomial of degree divisible by p whose highest-degree homogeneous form defines a non-singular projective hypersurface, and in the second case, f is a polynomial of degree prime to p whose highest-degree homogeneous form defines a projective hypersurface with isolated singularities.