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

Author

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

Keywords

• nonn

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).

Examples

			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)
		

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.

Crossrefs

Cf. A061255.

Programs

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]

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].