A361721 - OEIS

%I #25 May 10 2024 08:50:04

%S 1,2,3,5,8,13,20,31,47,70,103,151,218,313,446,629,883,1233,1711,2362,

%T 3244,4433,6034,8179,11043,14852,19906,26589,35400,46986,62182,82057,

%U 107989,141744,185583,242387,315842,410627,532687,689573,890837,1148567,1478020,1898430,2434006,3115202,3980232

%N 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).

%C 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.

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

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

%H Robin Visser, <a href="/A361721/b361721.txt">Table of n, a(n) for n = 0..200</a>

%H Y. W. Ding and Y. Ouyang, <a href="https://doi.org/10.1007/s10114-012-0490-8">A simple proof of Dieudonné-Manin classification theorem</a>, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 8, 1553-1558.

%H S. Harashita, <a href="https://doi.org/10.1007/s00209-020-02504-w">Asymptotic formula of the number of Newton polygons</a>, Math. Z. 297 (2021), no. 1-2, 113-132.

%H M. Rapoport, <a href="http://www.numdam.org/book-part/SB_2001-2002__44__207_0/">On the Newton stratification</a>, Astérisque No. 290 (2003), Séminaire Bourbaki, Exp. No. 903, viii, 207-224.

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

%F 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].

%e 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.

%e For n = 2, the a(2) = 3 possible symmetric Newton polygons of length 4 and depth 2 are:

%e  (0,0)--(4,2)

%e  (0,0)--(2,0)--(4,2)

%e  (0,0)--(1,0)--(3,1)--(4,2)

%e For n = 3, the a(3) = 5 possible symmetric Newton polygons of length 6 and depth 3 are:

%e  (0,0)--(6,3)

%e  (0,0)--(3,0)--(6,3)

%e  (0,0)--(3,1)--(6,3)

%e  (0,0)--(2,0)--(4,1)--(6,3)

%e  (0,0)--(1,0)--(5,2)--(6,3)

%o (Sage)  # Use generating function to return a(n)

%o def a(n):

%o     f = product([(1 - x^k)^(-euler_phi(k)) for k in range(1,n+1)])

%o     gf = sqrt((1+x)*f)/(1-x)

%o     return gf.taylor(x,0,n).coefficients()[n][0]

%Y Cf. A061255.

%K nonn

%O 0,2

%A Steven Groen, James Rawson, and _Robin Visser_, Mar 21 2023