OFFSET
0,3
COMMENTS
Let the random variable X be the coefficient of t^3 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].
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^3 in L_p(t/sqrt(p)), as p varies.
LINKS
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
FORMULA
See Prop. 12 of Kedlaya-Sutherland.
EXAMPLE
a(4) = 23 because E[X^4] = 23 for X the t^3 coeff of det(tI-A) in USp(6).
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew V. Sutherland, Mar 24 2008
STATUS
approved