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Thursday, October 31, 2019

Level 4

I. Moonshine function Define, a(τ)=(d32d1d24)8,b(τ)=(d1d4)8+8 then (d32d1d24)88=(d1d4)8+8 Equivalently, using the reciprocal of a(τ), (2d1d24d32)8+(d21d4d32)8=1 which are just versions of the single trinomial identity for level 4. We also have the four level 4 moonshine functions, j4A(τ)=((d1d4)4+16(d4d1)4)2=(d22d1d4)24j4B(τ)=((d2d4d1d8)44(d1d8d2d4)4)2=j2A(2τ)=j4D(τ)+26j4D(τ)j4C(τ)=(d1d4)8+8=b(τ)j4D(τ)=(d2d4)12 Note that the function j4C(τ)=16λ(2τ)8=(d1d4)8+8 with modular lambda function λ(τ) is a normalized Hauptmodul.

II. Comparisons: Notice that,j4C=(d32d1d24)88,j8E=(d34d2d28)4,j16B=(d38d4d216)2The last two will play analogous roles for levels 8 and 16. Also, j4A is remarkably a 24th power. Versions appear in levels that are 4m divisors of 24, j4A=(d22d1d4)24,j8B=(d24d2d8)12,j12D=(d26d3d12)8,j24E=(d212d6d24)4 Similarly, versions of j2B appear in levels that are 2m divisors of 24, j2B(τ)=(d1d2)24,j4D(τ)=(d2d4)12,j6F(τ)=(d3d6)8,j8F(τ)=(d4d8)6,j12J(τ)=(d6d12)4,j24J(τ)=(d12d24)2

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