cp's OEIS Frontend

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

%I A130542 #8 Oct 22 2023 00:26:09
%S A130542 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 A130542 24,11,8,18,16,20,12,12,16,21,12,12,30,13,12,24,18,13,12,18,22,24,14,
%U A130542 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 A130542 The maximum absolute value of the L-series coefficient for an elliptic curve.
%C A130542 The values of a(n) and the multiplicativity are conjectural.
%C A130542 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 A130542 Robin Visser, <a href="/A130542/b130542.txt">Table of n, a(n) for n = 1..10000</a>
%H A130542 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 A130542 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 A130542 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 A130542 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 A130542 (Sage)
%o A130542 def a(n):
%o A130542     ans, fcts = 1, Integer(n).factor()
%o A130542     for pp in fcts:
%o A130542         max_ap = 1
%o A130542         for ap in range(-floor(2*sqrt(pp[0])), floor(2*sqrt(pp[0]))+1):
%o A130542             app = [1, ap]
%o A130542             for i in range(pp[1]-1): app.append(app[1]*app[-1]-pp[0]*app[-2])
%o A130542             max_ap = max(max_ap, abs(app[-1]))
%o A130542         ans *= max_ap
%o A130542     return ans  # _Robin Visser_, Oct 21 2023
%K A130542 nonn,mult
%O A130542 1,2
%A A130542 _Michael Somos_, Jun 04 2007