A Borcherds Product in S2(K(523))+

(Filename: BP-2-523-1-0-1-hs.html)

DEFINITION OF ψ AND BORCH(ψ)
The weakly holomorphic Jacobi form ψ of weight 0 and index 523, is defined as
    ψ = Σj=152 cj (φj|V2)/φj
where the coefficients cj and theta blocks φj are given as follows. Note each φj ∈ J2,523cusp with ν(φj) = 1.
cjφj
2
TB(2;1,3,4,6,9,12,13,13,14,15)
-2
TB(2;1,1,2,4,11,12,13,13,14,15)
1
TB(2;1,2,4,8,10,11,12,12,14,16)
1
TB(2;1,2,5,7,9,11,12,13,14,16)
-2
TB(2;1,2,3,5,11,11,12,13,14,16)
-2
TB(2;2,3,4,5,9,11,13,13,14,16)
3
TB(2;1,2,7,7,8,9,11,14,15,16)
-1
TB(2;1,3,4,5,7,10,13,14,15,16)
1
TB(2;1,2,3,4,7,11,13,14,15,16)
-1
TB(2;1,3,6,7,8,9,10,15,15,16)
-1
TB(2;1,3,4,8,9,10,11,13,14,17)
2
TB(2;1,4,5,6,7,11,12,13,14,17)
1
TB(2;1,2,3,4,4,11,13,14,15,17)
-1
TB(2;1,4,5,7,8,9,11,12,16,17)
-2
TB(2;1,5,5,6,7,10,11,12,16,17)
-3
TB(2;3,3,4,7,7,10,10,13,16,17)
-1
TB(2;1,5,6,6,7,8,11,13,16,17)
1
TB(2;2,4,5,6,7,9,11,13,16,17)
-2
TB(2;1,3,5,6,7,8,11,14,16,17)
1
TB(2;1,1,2,3,5,11,12,14,16,17)
2
TB(2;1,1,4,7,8,8,9,15,16,17)
3
TB(2;3,3,4,7,7,10,11,12,15,18)
2
TB(2;1,3,5,6,7,8,12,13,15,18)
1
TB(2;1,2,4,6,7,8,10,14,16,18)
-2
TB(2;1,3,4,5,7,8,10,13,17,18)
3
TB(2;1,2,7,8,9,10,11,11,12,19)
-1
TB(2;2,3,5,7,8,10,11,12,13,19)
-2
TB(2;2,2,4,6,8,10,11,12,14,19)
-1
TB(2;1,5,6,7,8,8,9,13,14,19)
2
TB(2;3,4,5,6,7,8,11,13,14,19)
4
TB(2;2,3,5,5,7,8,12,13,14,19)
-2
TB(2;3,4,4,7,8,8,11,11,15,19)
-1
TB(2;1,4,4,5,5,9,10,14,15,19)
-2
TB(2;1,4,5,6,7,9,10,11,16,19)
2
TB(2;1,2,3,7,8,9,10,11,16,19)
-1
TB(2;3,4,4,5,7,7,11,12,16,19)
1
TB(2;1,3,4,5,5,6,11,14,16,19)
-2
TB(2;1,2,3,4,5,8,9,14,17,19)
-2
TB(2;1,1,3,4,7,8,10,11,18,19)
-2
TB(2;1,3,4,7,9,10,10,11,13,20)
1
TB(2;1,3,6,6,7,9,11,12,13,20)
1
TB(2;2,2,5,7,7,9,11,12,13,20)
-3
TB(2;1,1,4,7,8,9,11,12,13,20)
1
TB(2;2,3,5,6,7,8,11,13,13,20)
-1
TB(2;1,3,4,5,7,8,12,13,13,20)
-1
TB(2;2,3,4,6,8,10,10,11,14,20)
-1
TB(2;1,3,4,6,7,7,11,13,14,20)
1
TB(2;1,2,4,4,6,8,12,13,14,20)
-1
TB(2;1,3,4,5,7,10,10,11,15,20)
1
TB(2;1,4,4,6,7,8,8,12,16,20)
2
TB(2;1,2,3,7,7,8,9,10,17,20)
2
TB(2;1,3,4,5,6,7,10,11,17,20)
Note that BORCH(ψ) has leading Jacobi coefficient the theta block φ ∈ J2,523cusp, with ν(φ) = 1,
     φ = TB(2;2,2,4,5,−6,7,−8,9,9,11,12,13,14,16)
= 2241516−1718−192111121131141161
= η(τ)4(θ(τ,2z)/η(τ))2(θ(τ,4z)/η(τ))(θ(τ,5z)/η(τ))(θ(τ,6z)/η(τ))−1(θ(τ,7z)/η(τ))(θ(τ,8z)/η(τ))−1(θ(τ,9z)/η(τ))2(θ(τ,11z)/η(τ))(θ(τ,12z)/η(τ))(θ(τ,13z)/η(τ))(θ(τ,14z)/η(τ))(θ(τ,16z)/η(τ))
and so BORCH(ψ) begins with Jacobi coefficient #1.

PROPERTIES OF ψ AND BORCH(ψ)
See this page for properties of ψ and BORCH(ψ), such as Humbert divisors of BORCH(ψ) and some Fourier coefficients of BORCH(ψ).