Partitions into Distinct Parts and Elliptic Curves

Let Q(N) denote the number of partitions of N into distinct parts. If ω(k) := 3k2 +k 2 , then it is well known that Q(N) + ∞X k=1 (−1)k “ Q(N − 2ω(k)) + Q(N − 2ω(−k)) ” = ( 1 if N = m(m+1) 2 0 otherwise. 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. 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 qD in a certain power series is non-zero, and assuming t...

• Type: Journal
• Year: 1998
• Where: JCT
• Authors: Ken Ono
• Keywords: Distinct Parts, JCT 1998, Recurrence Type Relations, ‘congruent Number