OFFSET
1,2
COMMENTS
The values of a(n) and the multiplicativity are conjectural.
Let p be a prime number. By a theorem of Deuring and Waterhouse, for any integer t of absolute value at most floor(2*sqrt(p)), there exists an elliptic curve E having its p-th L-series coefficient as t. This gives the values a(n) for all primes and prime powers n. Multiplicativity of a(n) can be shown by an application of the Chinese remainder theorem for elliptic curves, thus yielding all values of a(n). - Robin Visser, Oct 21 2023
LINKS
Robin Visser, Table of n, a(n) for n = 1..10000
Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.
W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.
FORMULA
For primes p : a(p) = floor(2*sqrt(p)) and a(p^2) = floor(2*sqrt(p))^2 - p [Deuring-Waterhouse]. - Robin Visser, Oct 21 2023
EXAMPLE
PROG
(Sage)
def a(n):
ans, fcts = 1, Integer(n).factor()
for pp in fcts:
max_ap = 1
for ap in range(-floor(2*sqrt(pp[0])), floor(2*sqrt(pp[0]))+1):
app = [1, ap]
for i in range(pp[1]-1): app.append(app[1]*app[-1]-pp[0]*app[-2])
max_ap = max(max_ap, abs(app[-1]))
ans *= max_ap
return ans # Robin Visser, Oct 21 2023
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Michael Somos, Jun 04 2007
STATUS
approved