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A228073 Degrees of special cubic divisors C_i. Each C_i corresponds to the collection of cubic fourfolds in the complex projective space of dimension 5, and these cubic fourfolds are indexed by i (i.e., of degree i). For each index i, the degree can be interpreted as the number of such cubic fourfolds. 0

%I

%S -2,0,0,192,3402,0,196272,917568,0,11327232,32453136,0,200946468,

%T 447835392,0,1873218816,3568293162,0,11587373280,19916839872,0,

%U 54185699328,86195929680,0,205762215024,309802123008,0,668839597248,961259741172,0,1916372334240,2662571684160,0,4975266781440,6699605038272,0,11865484014132,15620739902208,0,26444647324032,34029871628112,0,55430090586720,70205139813312,0,110613834846720,137669694779232,0

%N Degrees of special cubic divisors C_i. Each C_i corresponds to the collection of cubic fourfolds in the complex projective space of dimension 5, and these cubic fourfolds are indexed by i (i.e., of degree i). For each index i, the degree can be interpreted as the number of such cubic fourfolds.

%C In the formula, alpha(q) and beta(q) are level three modular forms of weight 1 and 3 that generate the space of modular forms with respect to the group Gamma_0(3). Legendre(p/q) is the Legendre symbol.

%H B. Hassett, <a href="https://www.math.brown.edu/~bhassett/papers/cubics/cubiclong.pdf">Special Cubic Fourfolds</a>, Compositio Mathematica 120 (2000), no 1, 1-23.

%H L. Zhang and Z. Li, <a href="https://doi.org/10.1016/j.aim.2013.06.003">Modular Forms and Special Cubic Fourfolds</a>, Advances in Mathematics, 2013, 315-326.

%H L. Zhang and Z. Li, <a href="https://arxiv.org/abs/1203.1373">Modular Forms and Special Cubic Fourfolds</a>, arXiv:1203.1373 [math.AG], 2012-2015.

%F a(n) = coefficient of q^n in (-alpha3^11 + 162*alpha3^8*beta3 + 91854*alpha3^5*beta3^2 + 2204496*alpha3^2*beta3^3 - alpha^11 + 66*alpha^8*beta - 1386*alpha^5*beta^2 + 9072*alpha^2*beta^3), where alpha(q) = 1+6*Sum_{n>=1}(q^n * Sum_{d|n}Legendre(d/3)); alpha3(q) = alpha(q^3); beta(q) = Sum_{n>=1}(q^n * Sum_{d|n}(n/d)^2*Legendre(d/3)); beta3(q) = beta(q^3).

%e For n = 0, a(0)= -2, corresponds to degree of special cubic fourfold of discriminant 0;

%e For n = 3, a(3) = 192, corresponds to degree of special cubic fourfold of discriminant 2*3 = 6;

%e For n = 4, a(4) = 3402, corresponds to degree of special cubic fourfold of discriminant 2*4 = 8;

%e In general, a(n) != 0 if and only if 2n==0,2 mod 6.

%K sign

%O 0,1

%A _Letao Zhang_, Aug 08 2013

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