We obtain new bounds on short Weil sums over small multiplicative subgroups of prime finite fields which remain nontrivial in the range the classical Weil bound is already trivial. The method we use is a blend of techniques coming from algebraic geometry and additive combinatorics.

We prove general Dwork-type congruences for constant terms attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and p-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of hypergeometric and Knizhnik–Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application, we show that the simplest example of a p-adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of its monodromy representation.

Let F be a system of polynomial equations in one or more variables with integer coefficients. We show that there exists a univariate polynomial $D \in \mathbb {Z}[x]$ such that F is solvable modulo p if and only if the equation $D(x) \equiv 0 \pmod {p}$ has a solution.

In [Gorodnik and Nevo, Counting lattice points, J. Reine Angew. Math. 663 (2012), 127–176] an effective solution of the lattice point counting problem in general domains in semisimple S-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action of G on G/Γ, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in [Nevo and Sarnak, Prime and almost prime integral points on principal homogeneous spaces, Acta Math. 205 (2010), 361–402] and use them to establish several useful consequences of this property, including:

1. (1) effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties;

2. (2) effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties;

3. (3) effective lower bounds on the number of almost prime points on symmetric varieties;

4. (4) effective upper bounds on almost prime solutions of congruences in homogeneous varieties.

We show that a system of r quadratic forms over a 𝔭-adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax–Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605–630] requires the characteristic to be large in terms of the degree of the field over ℚp. The proofs use a 𝔭-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry.

This paper is concerned with non-trivial solvability in $p$-adic integers of systems of two and three additive forms. Assuming that the congruence equation $a x^k + b y^k + c z^k \equiv d \,(\mbox{mod}\,p)$ has a solution with $xyz \not\equiv 0\,(\mbox{mod}\,p)$ we have proved that any system of two additive forms of odd degree $k$ with at least $6 k + 1$ variables, and any system of three additive forms of odd degree $k$ with at least $14 k + 1$ variables always has non-trivial $p$-adic solutions, provided $p$ does not divide $k$. The assumption of the solubility of the congruence equation above is guaranteed for example if $p > k^4$.

In the particular case of degree $k = 5$ we have proved the following results. Any system of two additive forms with at least $n$ variables always has non-trivial $p$-adic solutions provided $n \geq 31$ and $p > 101$ or $n \geq 36$ and $p > 11$. Furthermore any system of three additive forms with at least $n$ variables always has non-trivial $p$-adic solutions provided $n \geq 61$ and $p > 101$ or $n \geq 71$ and $p > 11$.

Let $K$ be a $p$-adic field, let $Z_{\Phi }(s,f)$, $s\,{\in}\,\mathbb{C}$, with Re$(s)\,{>}\,0$, be the Igusa local zeta function associated to $f(x)\,{=}\,(f_{1}(x),\ldots,f_{l}(x))\,{\in}\,[ K( x_{1},\ldots,x_{n})]^{l}$, and let $\Phi $ be a Schwartz–Bruhat function. The aim of this paper is to describe explicitly the poles of the meromorphic continuation of $Z_{\Phi }(s,f)$. Using resolution of singularities it is possible to express $Z_{\Phi }(s,f)$ as a finite sum of $p$-adic monomial integrals. These monomial integrals are computed explicitly by using techniques of toroidal geometry. In this way, an explicit list of the candidates for poles of $Z_{\Phi }(s,f)$ is obtained.

To a polynomial f over a non-archimedean local field K and a character χ of the group of units of the valuation ring of K one associates Igusa’s local zeta function Z (s, f, χ). In this paper, we study the local zeta function Z(s, f, χ) associated to a non-degenerate polynomial f, by using an approach based on the p-adic stationary phase formula and Néron p-desingularization. We give a small set of candidates for the poles of Z (s, f, χ) in terms of the Newton polyhedron Γ(f) of f. We also show that for almost all χ, the local zeta function Z(s, f, χ) is a polynomial in q−s whose degree is bounded by a constant independent of χ. Our second result is a description of the largest pole of Z(s, f, χtriv) in terms of Γ(f) when the distance between Γ(f) and the origin is at most one.