For a polynomial f(x) in (Zp∩Q)[x] of degree d ≥ 3 let L(f⊗Fp;T ) be the L function of the exponential sum of f mod p. Let NP(f ⊗ Fp) denote the Newton polygon of L(f⊗Fp; T ). Let HP(f) denote the Hodge polygon of f , which is the lower convex hull in R2 of the points (n, n(n+1) 2d ) for 0 ≤ n ≤ d−1. We prove that there is a Zariski dense subset U defined over Q in the space A of degree-d monic...

We study analytic Zariski structures from the point of view of non-elementary model theory. We show how to associate an abstract elementary class with a one-dimensional analytic Zariski structure and prove that the class is stable, quasi-minimal and homogeneous over models. We also demonstrate how Hrushovski’s predimension arises in this general context as a natural geometric notion and use it ...

The notion of an analytic Zariski structure was introduced in [1] by the author and N.Peatfield in a form slightly different from the one presented here. Analytic Zariski generalises the previously known notion of a Zariski structure (see [2] for one-dimensional case and [3], [4] for the general definition) mainly by dropping the requirement of Noetherianity and weakening the assumptions on the...

where f is a polynomial of degree e in n variables with coefficients in a finite field F of characteristic p, E is an overfield of F, and ψ is an additive character of F. It is interesting to ask what happens with sums S(E, f) when one varies E, or when one varies f . In order to talk precisely about the latter, we denote by P(e, n) the scheme parameterizing all polynomials in n variables of de...

Given a fixed integer n, we consider closed subgroups G of GLn(Zp), where p is sufficiently large in terms of n. Assuming that the Zariski closure of G in GLn has no toric part, we give a condition on the (mod p) reduction of G which guarantees that G is of bounded index in GLn(Zp) ∩ G(Qp). In [No], Nori considered a special class of subgroups of GLn(Fp), namely groups which are generated by el...

Denote by $H(d_1,d_2,d_3)$ the set of all homogeneous polynomial mappings $F=(f_1,f_2,f_3): \mathbb{C}^3 \to \mathbb{C}^3$, such that $\deg f_i=d_i$. We show if $\gcd(d_i,d_j) \leq 2$ for $1 i < j\leq 3$ and $\gcd(d_1,d_2,d_3)=1$, then there is a non-empty Zariski open subset $U \subset H(d_1,d_2,d_3)$ every mapping $F\in U$ map germ $(F,0)$ $\mathcal{A}$-finitely determined. Moreover, in this ...

Motivated by work of Zhang from the early ‘90s, Medvedev and Scanlon formulated the following conjecture. Let F be an algebraically closed field of characteristic 0 and let X be a quasiprojective variety defined over F endowed with a dominant rational self-map φ. Then there exists a point x ∈ X(F ) with Zariski dense orbit under φ if and only if φ preserves no nontrivial rational fibration, i.e...

We begin by reviewing various classical problems concerning the existence of primes or numbers with few prime factors as well as some of the key developments towards resolving these long standing questions. Then we put the theory in a natural and general geometric context of actions on affine n-space and indicate what can be established there. The methods used to develop a combinational sieve i...