A138356 Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(4).

1, 1, 2, 4, 10, 27, 82, 268, 940, 3476, 13448, 53968, 223412, 949535, 4128594, 18310972, 82645012, 378851428, 1760998280, 8288679056, 39457907128, 189784872428, 921472827272, 4512940614960, 22279014978544, 110797225212112

Offset: 0

Author

Andrew V. Sutherland, Mar 17 2008

Keywords

• nonn

Comments

Let the random variable X be the coefficient of t^2 in the characteristic polynomial det(tI-A) of a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic). Then a(n) = E[X^n].

Let L_p(T) be the L-polynomial (numerator of the zeta function) of a genus 2 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.

See A095922 for central moments.

Examples

			a(3) = 4 because E[X^3] = 4 for X the t^2 coeff of det(tI-A) in USp(4).
a(3) = 1*2^3*(1*1-0^2) + 3*2^2*(0*0-1^2) + 3*2^1*(1*2-0^2) + 1*2^0*(0*0-2^2) = 8 - 12 + 12 - 4 = 4.
		

Links

• Kiran S. Kedlaya, Andrew V. Sutherland, Computing L-series of hyperelliptic curves, arXiv:0801.2778 [math.NT], 2008-2012; Algorithmic Number Theory Symposium--ANTS VIII, 2008.
• Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
• Nicholas M. Katz and Peter Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy, AMS, 1999.

Crossrefs

Cf. A095922, A138349.

Formula

a(n) = (1/2)Integral_{x=0..Pi,y=0..Pi}(4cos(x)cos(y)+2)^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy.

a(n) = Sum_{i=0..n}binomial(n,i)2^{n-i}*(A126120(i)*A126120(i+2)-A126120(i+1)^2).