NOTE ON A PAPER OF BIRCH

A. O. L. ATKIN

1. The exponential sum2R

- VVo = 0 6 = 0

p

Z exp I (ax3 + bx) 1

0

which is integral for prime p , has been considered by Birch [1], who showed thatfor 1 ̂ R ^ 4 and p ^ 5 we have

where

£*(/>) = (2*-1)! ( / > - ! ) / C2/?-

For R = 5 we find

where Birch observes that y{p) is integral and \y(p)\< 2/?3/2, and suggested to theauthor that it should have an interpretation in terms of modular forms.

2. Let'

^(T) = eniT/l2 f l (1 -x1") (* = e2 n")

r = l

and

where ^ = A* and «(«) = 0 if n ^ 3 (mod 8) or « is non-integral. Then we have

a(np2)+p(5n/p)a(n)+p3 a(np~2) = k{p)a(n) if /? = 3 or p ^ 7,

where A(/?) is the eigenvalue of the Hecke operator T(p2) acting on F(x) and theequations hold for all integral n.

3. We now

Conjecture. For p ^ 5 we have y(p) = —(—There is agreement for p ^ 311.

1. Birch, B. J., How the number of points on an elliptic curve over a fixed prime field varies(to appear).

The Atlas Computer Laboratory,Chilton, Didcot, Berks.

Received 18 January, 1968

[J. LONDON MATH. SOC, 44 (1969), 282]