A130542 The maximum absolute value of the L-series coefficient for an elliptic curve.

1, 2, 3, 2, 4, 6, 5, 3, 6, 8, 6, 6, 7, 10, 12, 4, 8, 12, 8, 8, 15, 12, 9, 9, 11, 14, 9, 10, 10, 24, 11, 8, 18, 16, 20, 12, 12, 16, 21, 12, 12, 30, 13, 12, 24, 18, 13, 12, 18, 22, 24, 14, 14, 18, 24, 15, 24, 20, 15, 24, 15, 22, 30, 8, 28, 36, 16, 16, 27, 40, 16, 18, 17, 24, 33, 16, 30, 42

Offset: 1

Author

Michael Somos, Jun 04 2007

Keywords

• nonn
• mult

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

Examples

			For example abs(A007653(n)) <= a(n) for all n where A007653 is the L-series for the curve y^2 - y = x^3 - x.
		

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.

Programs

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

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