In mathematics, the Weber modular functions are a family of three modular functions f, f1, and f2, studied by Heinrich Martin Weber.
Let q = e 2 π i τ where τ is an element of the upper half-plane.
f ( τ ) = q − 1 48 ∏ n > 0 ( 1 + q n − 1 2 ) = e − π i 24 η ( τ + 1 2 ) η ( τ ) = η 2 ( τ ) η ( τ 2 ) η ( 2 τ ) f 1 ( τ ) = q − 1 48 ∏ n > 0 ( 1 − q n − 1 2 ) = η ( τ 2 ) η ( τ ) f 2 ( τ ) = 2 q 1 24 ∏ n > 0 ( 1 + q n ) = 2 η ( 2 τ ) η ( τ ) where η ( τ ) is the Dedekind eta function. Note the η quotients immediately imply that
f ( τ ) f 1 ( τ ) f 2 ( τ ) = 2 . The transformation τ → –1/τ fixes f and exchanges f1 and f2. So the 3-dimensional complex vector space with basis f, f1 and f2 is acted on by the group SL2(Z).
Let the argument of the Jacobi theta function be the nome q = e π i τ . Then,
f ( τ ) = θ 3 ( 0 , q ) η ( τ ) f 1 ( τ ) = θ 4 ( 0 , q ) η ( τ ) f 2 ( τ ) = θ 2 ( 0 , q ) η ( τ ) Thus,
f 1 ( τ ) 8 + f 2 ( τ ) 8 = f ( τ ) 8 which is simply a consequence of the well known identity,
θ 2 ( 0 , q ) 4 + θ 4 ( 0 , q ) 4 = θ 3 ( 0 , q ) 4 The three roots of the cubic equation,
j ( τ ) = ( x + 16 ) 3 x where j(τ) is the j-function are given by x i = f ( τ ) 24 , f 1 ( τ ) 24 , f 2 ( τ ) 24 . Also, since,
j ( τ ) = 32 ( θ 2 ( 0 , q ) 8 + θ 3 ( 0 , q ) 8 + θ 4 ( 0 , q ) 8 ) 3 ( θ 2 ( 0 , q ) θ 3 ( 0 , q ) θ 4 ( 0 , q ) ) 8 then,
j ( τ ) = ( f ( τ ) 16 + f 1 ( τ ) 16 + f 2 ( τ ) 16 2 ) 3 