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Publication# 2-Selmer groups and the Birch-Swinnerton-Dyer Conjecture for the congruent number curves

Abstract

We take an approach toward Counting the number of integers n for which the curve (n),: y(2) = x(3) - n(2)x has 2-Selmer groups of a given size. This question was also discussed in a pair of papers by Roger Heath-Brown. In contrast to earlier work, our analysis focuses oil restricting the number of prime factors of n. Additionally, we discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the "independence" of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors. (C) 2009 Elsevier Inc. All rights reserved.

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Related concepts (29)

Related publications (32)

Ontological neighbourhood

Prime number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4.

Divisor

In mathematics, a divisor of an integer , also called a factor of , is an integer that may be multiplied by some integer to produce . In this case, one also says that is a multiple of An integer is divisible or evenly divisible by another integer if is a divisor of ; this implies dividing by leaves no remainder. An integer n is divisible by a nonzero integer m if there exists an integer k such that . This is written as Other ways of saying the same thing are that m divides n, m is a divisor of n, m is a factor of n, and n is a multiple of m.

Zero divisor

In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0, or equivalently if the map from R to R that sends x to ax is not injective. Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y in R such that ya = 0. This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor.

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