Partitions into Distinct Parts and Elliptic Curves

✍ Scribed by Ken Ono

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 168 KB
• Volume: 82
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1997.2847

✦ Synopsis

Let Q(N ) denote the number of partitions of N into distinct parts. If |(k) := (3k 2 +k)Â2, then it is well known that

In this short note we start with Tunnell's work on the congruent number problem'' and show that Q(N ) often satisfies weighted'' recurrence type relations. For every N there is a relation for Q(N ) which may involve a special value of an elliptic curve L-function.

1998 Academic Press

A positive integer D is called a ``congruent number'' if there exists a right triangle with rational sidelengths with area D. Over the centuries there have been many investigations attempting to classify the congruent numbers, but little was known until Tunnell [T] brilliantly applied a tour de force of methods and provided a conditional solution to this problem. It turns out that a square-free integer D is not congruent if the coefficient of q D in a certain power series is non-zero, and assuming the Birch and Swinnerton Dyer Conjecture D is congruent if the coefficient of q D is zero.

In this note we start with Tunnell's work and obtain weighted recurrence formulas for Q(N ), the number of partitions into distinct parts. We begin by defining the critical objects. Define integers b(n) by the infinite product : n=1 b(n) q n :=q ` n=1 (1&q 4n ) 2 (1&q 8n ) 2 .