Power-free values, large deviations, and integer points on irrational curves

Let $f\in \mathbb{Z}\lbrack x\rbrack$ be a polynomial of degree $d\geq 3$ without roots of multiplicity $d$ or $(d-1)$. Erd\H{o}s conjectured that, if $f$ satisfies the necessary local conditions, then $f(p)$ is free of $(d-1)$th powers for infinitely many primes $p$. This is proved here for all $f$ with sufficiently high entropy. The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov's theorem from the theory of large deviations.Comment: 39 pages; rather major revision, with strengthened and generalized statement

Root numbers and ranks in positive characteristic

For a global field K and an elliptic curve E_eta over K(T), Silverman's specialization theorem implies that rank(E_eta(K(T))) <= rank(E_t(K)) for all but finitely many t in P^1(K). If this inequality is strict for all but finitely many t, the elliptic curve E_eta is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial. Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K = kappa(u) over any finite field kappa with odd characteristic, we construct an explicit 2-parameter family E_{c,d} of non-isotrivial elliptic curves over K(T) (depending on arbitrary c, d in kappa^*) such that, under the parity conjecture, each E_{c,d} has elevated rank.Comment: 40 pages; last version; to appear in Adv. Mat