Let X be a smooth quasiprojective subscheme of Pn of dimension m≥ 0 over Fq. Then there
exist homogeneous polynomials f over Fq for which the intersection of X and the …

Diophantine number theory is an active area that has seen tremendous growth over the past
century, and in this theory unit equations play a central role. This comprehensive treatment …

We present a heuristic that suggests that ranks of elliptic curves E over Q are bounded. In
fact, it suggests that there are only finitely many E of rank greater than 21. Our heuristic is …

Following Katz–Sarnak, Iwaniec–Luo–Sarnak and Rubinstein, we use the one-and two-level
densities to study the distribution of low-lying zeros for one-parameter rational families of …

We give an upper bound on the number of extensions of a fixed number field of prescribed
degree and discriminant≤ X; these bounds improve on work of Schmidt. We also prove …

Given f∈Zx\sb1,...x\sbn, we compute the density of x∈Z\spn such that f(x) is squarefree,
assuming the abc-conjecture. Given f,g∈Zx\sb1,...x\sbn, we compute unconditionally the …

We determine the density of monic integer polynomials of given degree n> 1 that have
squarefree discriminant; in particular, we prove for the first time that the lower density of such …

We define a notion of height for rational points with respect to a vector bundle on a proper
algebraic stack with finite diagonal over a global field, which generalizes the usual notion for …

We develop a method for determining the density of squarefree values taken by certain
multivariate integer polynomials that are invariants for the action of an algebraic group on a …

We define a notion of height for rational points with respect to a vector bundle on a proper
algebraic stack with finite diagonal over a global field, which generalizes the usual notion for …