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%I #8 Oct 22 2023 00:26:09
%S 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,
%T 24,11,8,18,16,20,12,12,16,21,12,12,30,13,12,24,18,13,12,18,22,24,14,
%U 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
%N The maximum absolute value of the L-series coefficient for an elliptic curve.
%C The values of a(n) and the multiplicativity are conjectural.
%C 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
%H Robin Visser, <a href="/A130542/b130542.txt">Table of n, a(n) for n = 1..10000</a>
%H Max Deuring, <a href="https://doi.org/10.1007/BF02940746">Die Typen der Multiplikatorenringe elliptischer Funktionenkörper</a>, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.
%H W. C. Waterhouse, <a href="https://doi.org/10.24033/asens.1183">Abelian varieties over finite fields</a>, Ann Sci. E.N.S., (4) 2 (1969), 521-560.
%F 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
%e 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.
%o (Sage)
%o def a(n):
%o ans, fcts = 1, Integer(n).factor()
%o for pp in fcts:
%o max_ap = 1
%o for ap in range(-floor(2*sqrt(pp[0])), floor(2*sqrt(pp[0]))+1):
%o app = [1, ap]
%o for i in range(pp[1]-1): app.append(app[1]*app[-1]-pp[0]*app[-2])
%o max_ap = max(max_ap, abs(app[-1]))
%o ans *= max_ap
%o return ans # _Robin Visser_, Oct 21 2023
%K nonn,mult
%O 1,2
%A _Michael Somos_, Jun 04 2007