A138550 -id:A138550 - cp's OEIS frontend

A138549 Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(6).

1, 1, 2, 5, 16, 62, 282, 1459, 8375, 52323, 350676, 2493846, 18659787, 145918295, 1186129168, 9978055080, 86545684565, 771571356565, 7051538798490, 65913863945775, 628919704903746, 6114899366942556, 60492393411513722

Offset: 0

Andrew V. Sutherland, Mar 24 2008

nonn

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].
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.
See A138550 for central moments.

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

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.

Cf. A138540, A138550, A138356.

See Prop. 12 of first Kedlaya-Sutherland reference.

A251591 Dimension of space of invariant tensors in 2n-th tensor power of the third fundamental representation of Sp(6).

Original entry on oeis.org

1, 1, 4, 35, 560, 14973, 589743, 30078048, 1824041570, 125400975830, 9507019477382, 78070828079199, 68560287232877740, 6376178095301876005, 623169409884847073730, 636070059202675270255520, 6745818886029778590765570, 740194253157571009569356970

Offset: 0

Bruce Westbury, Dec 05 2014

Cf. A138540, A138550.

LiE
```
p_tensor(2*n,[0,0,1],C3)|[0,0,0]
```

cp's OEIS Frontend

A138549 Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(6).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A251591 Dimension of space of invariant tensors in 2n-th tensor power of the third fundamental representation of Sp(6).

Views

Author

Keywords

Crossrefs

Programs

LiE