We study, in the case of ordinary primes, some connections between the GL2 and cyclotomic Iwasawa theory of an elliptic curve without complex multiplication. 2000 Mathematics Subject Classification: 11G05; 11R23.

This is a survey article on the Hardy-Littlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture. Mathematics Subject Classification (2000): 11L07, 11L20, 11L40, 11N13, 11N32, 11N37

We show that the parity conjecture for Selmer groups is invariant under deformation in p-adic families of self-dual pure Galois representations satisfying Pančǐskin’s condition at all primes above p. 2000 Mathematics Subject Classification: 11G40 11R23

This paper is concerned with the number of primes in short intervals. We present a method to use mean value estimates for the number of primes in (x, x+x] to obtain the asymptotic behavior of ψ(x+x)−ψ(x). The main idea is to use the properties of the exceptional set for the distribution of primes in short intervals. Mathematics Subject Classification (2000). 11NO5.

ABSTRACT. Let R be a commutative noetherian ring, I and J are two ideals of R. Inthis paper we introduce the concept of (I;J)- minimax R- module, and it is shown thatif M is an (I;J)- minimax R- module and t a non-negative integer such that HiI;J(M) is(I;J)- minimax for all i

Let E/Q be a modular elliptic curve, and p > 3 a good ordinary or semistable prime. Under mild hypotheses, we prove an exact formula for the μ-invariant associated to the weight-deformation of the Tate module of E. For example, at ordinary primes in the range 3 < p < 100, the result implies the triviality of the μ-invariant of X0(11). 2000 Mathematics Subject Classification: 11G40; also 11F33, ...

The Hardy-Littlewood method is a well-known technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3-term progressions of primes, all less than N . This article surveys recent developments of the...

Enumeration of the twin primes, and the sum of. their reciprocals, is extended to 3 x 1015 , yielding the count 1r2 (3 x 1015 ) = 3310517800844. A more accurate estimate is obtained for Brun 's constant, B2 = 1.90216 05823 ± 0.00000 00008. Error analysis is presented to support the contention that this estimate produces a 95 % confidence interval for B2 • In addition, published values of the co...

The middle-third Cantor set C3 is a fractal consisting of all the points in [0, 1] which have non-terminating base-3 representations involving only the digits 0 and 2. I prove that all prime numbers p > 3 whose reciprocals belong to C3 must be base-3 repunit primes, and, conversely, that the reciprocals of all base-3 repunit primes must be in C3. This one-one correspondence appears to be unique...

We investigate the Galois group GS(p) of the maximal pextension unramified outside a finite set S of primes of a number field in the (tame) case, when no prime dividing p is in S. We show that the cohomology of GS(p) is ‘often’ isomorphic to the étale cohomology of the scheme Spec(Ok rS), in particular, GS(p) is of cohomological dimension 2 then. 2000 Mathematics Subject Classification: 11R34, ...