This talk reports on joint work with O. Ramaré where it is shown that if the sequence of prime numbers is coloured with K colours then every sufficiently large integer admits a representation as a sum of no more than CKlog log 4K monochromatic primes, where C is an absolute constant. This is deduced from a upper bound for the additive energy of dense sets of primes.
Arithmetic over the integers (or in number fields) is in many ways similar to arithmetic with polynomials in one variable over a finite field (or in function fields). In 1935 Leonard Carlitz introduced a remarkable function field analogue of the usual logarithm (which was later rediscovered and generalized by Drinfeld). In this talk I will explain what this "Carlitz logarithm" is, and how it can be used to obtain a function field version of the analytic class number formula. Although the statement is surprisingly similar to the classical formula, the only proof I know is by very different, more algebraic, methods. ( ref )
We study the number of rational points with bounded heights in a neighborhood of a plane curve. We will describe structural features of this set of rational points, from which new bounds for the number of such points will be deduced.
Let K be a number field, and let E be an elliptic curve over K. A famous result by Faltings of 1983 can be reformulated for elliptic curves as follows: if S is a set of primes of good reduction for E having density one, then the K-isogeny class of E is determined by the function which maps a prime p in S to the size #E(k_p) of the group of points over the residue field. We prove that it suffices to look at the radical of the size. We also replace E(k_p) by the image of the Mordell-Weil group via the reduction modulo p, and solve this analogue problem for a large class of abelian varieties. This is a joint work with Chris Hall.
(33) 3 20 43 68 75
: (32) 16 32 70 06