228 research outputs found
Power-free values, large deviations, and integer points on irrational curves
Let be a polynomial of degree
without roots of multiplicity or . Erd\H{o}s conjectured that, if
satisfies the necessary local conditions, then is free of th
powers for infinitely many primes . This is proved here for all 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
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