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Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(6).
2

%I #10 May 10 2019 04:33:20

%S 1,1,2,5,16,62,282,1459,8375,52323,350676,2493846,18659787,145918295,

%T 1186129168,9978055080,86545684565,771571356565,7051538798490,

%U 65913863945775,628919704903746,6114899366942556,60492393411513722

%N Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(6).

%C Let the random variable X be the coefficient of t^2 in the characteristic polynomial det(tI-A) of a random matrix in USp(6) (6x6 complex matrices that are unitary and symplectic). Then a(n) = E[X^n].

%C Let L_p(T) be the L-polynomial (numerator of the zeta function) of a genus 3 curve C. Under a generalized Sato-Tate conjecture, for almost all C, a(n) is the n-th moment of the coefficient of t^2 in L_p(t/sqrt(p)), as p varies.

%C See A138550 for central moments.

%H Kiran S. Kedlaya, Andrew V. Sutherland, <a href="https://arxiv.org/abs/0801.2778">Computing L-series of hyperelliptic curves</a>, arXiv:0801.2778 [math.NT], 2008-2012; Algorithmic Number Theory Symposium--ANTS VIII, 2008.

%H Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</a>, arXiv:0803.4462 [math.NT], 2008-2010.

%H Nicholas M. Katz and Peter Sarnak, <a href="http://bookstore.ams.org/coll-45/">Random Matrices, Frobenius Eigenvalues and Monodromy</a>, AMS, 1999.

%F See Prop. 12 of first Kedlaya-Sutherland reference.

%e a(3) = 5 because E[X^3] = 5 for X the t^2 coeff of det(tI-A) in USp(6).

%Y Cf. A138540, A138550, A138356.

%K nonn

%O 0,3

%A _Andrew V. Sutherland_, Mar 24 2008