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

1,2

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

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.

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

Sequence in context: A062068 A328219 A328879 * A128502 A349382 A244306

Adjacent sequences: A130539 A130540 A130541 * A130543 A130544 A130545

Keyword

nonn,mult

Author

Michael Somos, Jun 04 2007

Status

approved