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
KEYWORD
nonn
AUTHOR
Steven Groen, James Rawson, and Robin Visser, Mar 21 2023
STATUS
approved