In this expository note, we describe an arithmetic pairing associated to an isogeny between Abelian varieties over a finite field. We show that it generalises the Frey–Rück pairing, thereby giving a short proof of the perfectness of the latter.
Let Vₙ(P,Q) denote the generalized Lucas sequence with parameters P and Q. For all odd relatively prime values of P and Q such that P² + 4Q > 0, we determine all indices n such that Vₙ(P,Q) = 7kx² when k|P. As an application, we determine all indices n such that the equation Vₙ = 21x² has solutions.
The object of this paper is to present new proofs of the classical ternary theorems of additive prime number theory. Of these the best known is Vinogradov's result on the representation of odd numbers as the sums of three primes; other results will be discussed later. Earlier treatments of these problems used the Hardy-Littlewood circle method, and are highly analytical. In contrast, the method we use here is a (technically) elementary deduction from the Siegel-Walfisz Prime Number Theory. It uses...
For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and It is known that . Let ε > 0 be arbitrary and . We prove that for all positive integers r ≤ R, with at most exceptions, the Diophantine equation ⎧N = p₁+p₂+p₃, ⎨ j = 1,2,3,⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.
We determine explicitly the structure of the torsion group over the maximal abelian extension of and over the maximal -cyclotomic extensions of for the family of rational elliptic curves given by , where is an integer.