A361721 a(n) = number of isogeny classes of p-divisible groups of abelian varieties of dimension n over an algebraically closed field of characteristic p (for any fixed prime p).

1, 2, 3, 5, 8, 13, 20, 31, 47, 70, 103, 151, 218, 313, 446, 629, 883, 1233, 1711, 2362, 3244, 4433, 6034, 8179, 11043, 14852, 19906, 26589, 35400, 46986, 62182, 82057, 107989, 141744, 185583, 242387, 315842, 410627, 532687, 689573, 890837, 1148567, 1478020, 1898430, 2434006, 3115202, 3980232

Offset

0,2

Comments

a(n) is the number of p-divisible groups (also called Barsotti-Tate groups) of height 2n which are isomorphic to their own Cartier dual.

The Dieudonné-Manin classification theorem proves that a(n) is the number of symmetric Newton polygons of height 2n and depth n.

S. Harashita proved that log(a(n)) ~ (3/2)*(zeta(3)/zeta(2))^(1/3) * n^(2/3).

Links

Robin Visser, Table of n, a(n) for n = 0..200

Y. W. Ding and Y. Ouyang, A simple proof of Dieudonné-Manin classification theorem, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 8, 1553-1558.

S. Harashita, Asymptotic formula of the number of Newton polygons, Math. Z. 297 (2021), no. 1-2, 113-132.

M. Rapoport, On the Newton stratification, Astérisque No. 290 (2003), Séminaire Bourbaki, Exp. No. 903, viii, 207-224.

Formula

G.f.: sqrt((1+x)*f(x))/(1-x) where f(x) = Product_{k>=1} (1 - x^k)^(-phi(k)).

a(n) ~ 2*K^(1/2) / (sqrt(6*Pi) * C^(7/36) * (2*n)^(11/36)) * exp((3/4)*C^(1/3) * (2n)^(2/3) + (1/2)*(Sum_a g_a(C^(1/3) * (2n)^(-1/3)))), where C = 2*zeta(3)/zeta(2), K = exp(-2*zeta'(-1) - log(2*Pi)/6), g_a(x) is the residue of Gamma(s)*zeta(s+1)*zeta(s-1)/(zeta(s)*x^s) at s=a, and where Sum_a runs through all nontrivial zeros a of zeta(s) [Harashita].

Example

We denote a symmetric Newton polygon of height 2n and depth n as a sequence of nonnegative integer coordinates: (0,0)--(x1,y1)--(x2,y2)--...--(xk,yk)--(2n,n) such that the slope of the line through (xi, yi), (x_{i+1}, y_{i+1}) is strictly less than the slope of the line through (x_{i+1}, y_{i+1}), (x_{i+2}, y_{i+2}), and such that, for any 0 < x < 2n, the slope at x plus the slope at 2n-x equals 1.
For n = 2, the a(2) = 3 possible symmetric Newton polygons of length 4 and depth 2 are:
 (0,0)--(4,2)
 (0,0)--(2,0)--(4,2)
 (0,0)--(1,0)--(3,1)--(4,2)
For n = 3, the a(3) = 5 possible symmetric Newton polygons of length 6 and depth 3 are:
 (0,0)--(6,3)
 (0,0)--(3,0)--(6,3)
 (0,0)--(3,1)--(6,3)
 (0,0)--(2,0)--(4,1)--(6,3)
 (0,0)--(1,0)--(5,2)--(6,3)

Prog

(Sage)  # Use generating function to return a(n)
def a(n):
    f = product([(1 - x^k)^(-euler_phi(k)) for k in range(1, n+1)])
    gf = sqrt((1+x)*f)/(1-x)
    return gf.taylor(x, 0, n).coefficients()[n][0]

Crossrefs

Cf. A061255.

Sequence in context: A088795 A156145 A216053 * A173597 A059923 A266331

Adjacent sequences: A361718 A361719 A361720 * A361722 A361723 A361724

Keyword

nonn

Author

Steven Groen, James Rawson, and Robin Visser, Mar 21 2023

Status

approved