OFFSET
1,1
COMMENTS
E is singular over GF(p(5)) = GF(11) so we take p != 11.
Hasse proved that |a(n)| <= 2*sqrt(p) where p is p(n) or p(n+1) according as n < 5 or n >= 5.
Elkies proved that a(n) = 0 for infinitely many n.
REFERENCES
N. Koblitz, Introduction to Elliptic Curves and Modular Forms. New York: Springer-Verlag, 1993.
J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math., vol. 106, Springer-Verlag, Berlin and New York, 1986.
LINKS
Robin Visser, Table of n, a(n) for n = 1..10000
B. Mazur, The Structure of Error Terms in Number Theory and an Introduction to the Sato-Tate Conjecture, Current Events Bulletin, Amer. Math. Soc., 2007.
FORMULA
a(n) = -b(p) where q * Product_{k>=1} ((1 - q^k)*(1 - q^(11*k)))^2 = Sum_{k>=1} b(k)*q^k is the g.f. of A006571 and p is p(n) or p(n+1) according as n < 5 or n >= 5.
EXAMPLE
q*Product_{k>=1} ((1 - q^k)*(1 - q^11k))^2 = q - 2q^2 - ..., so a(1) = -b(p(1)) = -b(2) = -(-2) = 2.
PROG
(Sage)
def a(n):
if n < 5: p = Primes()[n-1]
else: p = Primes()[n]
E = EllipticCurve(GF(p), [0, -1, 1, 0, 0])
return -E.trace_of_frobenius() # Robin Visser, Jul 01 2023
CROSSREFS
KEYWORD
sign
AUTHOR
Jonathan Sondow, Jan 12 2007
EXTENSIONS
More terms from Robin Visser, Jul 01 2023
STATUS
approved