The Distribution of Galois Groups and Hilbert's Irreducibility Theorem

Density of rational points on cyclic covers of \mathbb{P}^n

“…If the degree of f is at least two, Serre [10] proved that N (f, B) B n+ 1 2 (log B) γ with γ < 1. Here the implied constant depends both on f and γ. Serre's proof is based on an analogous result proved by Cohen [3] in the affine case. The proof is a beautiful application of the large sieve inequalities.…”

Non-Archimedean Yomdin–gromov Parametrizations and Points of Bounded Height

“…Basically, the method consists in using a determinant of a suitable set of monomials evaluated at the integral points, in order to construct a family of auxiliary polynomials vanishing at all integral points on the curve within a small enough box. Building on the estimates in [2] for algebraic curves, Pila proved in [30] bounds on the number of integral (respectively, rational) points of bounded height on affine (respectively, projective) algebraic varieties of any dimension, improving on previous results by Cohen using the large sieve method [16]. Important further improvements going toward optimal bounds conjectured by Serre in [41, Section 13] have been made since by Heath-Brown et al [4,23,37].…”

“…When d > 1, Theorem 5.6.3 provides a nontrivial improvement on the 'trivial' bound with δ r m. Note however that, using the function field version of the large sieve inequality due to Hsu [24] instead of the one used in [41], one can easily adapt the arguments given in [41] to get the following function field analog of Cohen's result in [16]: if X is an irreducible subvariety of A n F q ((t)) of dimension m and degree d 2, then #X r = O(rq r (m−1/2) ), with X r := X (F q [t]) ∩ (F q [t] <r ) n .…”

The number of extensions of a number field with fixed degree and bounded discriminant

“…It is clear that the number of points on Z(Z) projecting to the box (2.7) is at most X 17/10 , but applying the large sieve to the map Z Π → A 5 (cf. [5] or [19]) one obtains the improved bound X 8/5+ε . (Note that the results, for example in [19], are stated only for a "square" box (all sides equal) around the origin -but indeed they apply, with uniform implicit constant, to a square box centered at any point.…”

“…Example 2.7. Let G = (1, 6, 2)(3, 4, 5), (5,6) (3, 4) ; it is a primitive permutation group on {1, 2, 3, 4, 5, 6} whose action is conjugate to the action of PSL 2 (F 5 ) on P 1 (F 5 ).…”