In mathematics, the Weber modular functions are a family of three functions f, f1, and f2, studied by Heinrich Martin Weber.
Definition
Let q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau }} where τ is an element of the upper half-plane. Then the Weber functions are
f ( τ ) = q − 1 48 ∏ n > 0 ( 1 + q n − 1 / 2 ) = η 2 ( τ ) η ( τ 2 ) η ( 2 τ ) = e − π i 24 η ( τ + 1 2 ) η ( 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 ) . {\displaystyle {\begin{aligned}{\mathfrak {f}}(\tau )&=q^{-{\frac {1}{48}}}\prod _{n>0}(1+q^{n-1/2})={\frac {\eta ^{2}(\tau )}{\eta {\big (}{\tfrac {\tau }{2}}{\big )}\eta (2\tau )}}=e^{-{\frac {\pi i}{24}}}{\frac {\eta {\big (}{\frac {\tau +1}{2}}{\big )}}{\eta (\tau )}},\\{\mathfrak {f}}_{1}(\tau )&=q^{-{\frac {1}{48}}}\prod _{n>0}(1-q^{n-1/2})={\frac {\eta {\big (}{\tfrac {\tau }{2}}{\big )}}{\eta (\tau )}},\\{\mathfrak {f}}_{2}(\tau )&={\sqrt {2}}\,q^{\frac {1}{24}}\prod _{n>0}(1+q^{n})={\frac {{\sqrt {2}}\,\eta (2\tau )}{\eta (\tau )}}.\end{aligned}}}These are also the definitions in Duke's paper "Continued Fractions and Modular Functions".2 The function η ( τ ) {\displaystyle \eta (\tau )} is the Dedekind eta function and ( e 2 π i τ ) α {\displaystyle (e^{2\pi i\tau })^{\alpha }} should be interpreted as e 2 π i τ α {\displaystyle e^{2\pi i\tau \alpha }} . The descriptions as η {\displaystyle \eta } quotients immediately imply
f ( τ ) f 1 ( τ ) f 2 ( τ ) = 2 . {\displaystyle {\mathfrak {f}}(\tau ){\mathfrak {f}}_{1}(\tau ){\mathfrak {f}}_{2}(\tau )={\sqrt {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).
Alternative infinite product
Alternatively, let q = e π i τ {\displaystyle q=e^{\pi i\tau }} be the nome,
f ( q ) = q − 1 24 ∏ n > 0 ( 1 + q 2 n − 1 ) = η 2 ( τ ) η ( τ 2 ) η ( 2 τ ) , f 1 ( q ) = q − 1 24 ∏ n > 0 ( 1 − q 2 n − 1 ) = η ( τ 2 ) η ( τ ) , f 2 ( q ) = 2 q 1 12 ∏ n > 0 ( 1 + q 2 n ) = 2 η ( 2 τ ) η ( τ ) . {\displaystyle {\begin{aligned}{\mathfrak {f}}(q)&=q^{-{\frac {1}{24}}}\prod _{n>0}(1+q^{2n-1})={\frac {\eta ^{2}(\tau )}{\eta {\big (}{\tfrac {\tau }{2}}{\big )}\eta (2\tau )}},\\{\mathfrak {f}}_{1}(q)&=q^{-{\frac {1}{24}}}\prod _{n>0}(1-q^{2n-1})={\frac {\eta {\big (}{\tfrac {\tau }{2}}{\big )}}{\eta (\tau )}},\\{\mathfrak {f}}_{2}(q)&={\sqrt {2}}\,q^{\frac {1}{12}}\prod _{n>0}(1+q^{2n})={\frac {{\sqrt {2}}\,\eta (2\tau )}{\eta (\tau )}}.\end{aligned}}}The form of the infinite product has slightly changed. But since the eta quotients remain the same, then f i ( τ ) = f i ( q ) {\displaystyle {\mathfrak {f}}_{i}(\tau )={\mathfrak {f}}_{i}(q)} as long as the second uses the nome q = e π i τ {\displaystyle q=e^{\pi i\tau }} . The utility of the second form is to show connections and consistent notation with the Ramanujan G- and g-functions and the Jacobi theta functions, both of which conventionally uses the nome.
Relation to the Ramanujan G and g functions
Still employing the nome q = e π i τ {\displaystyle q=e^{\pi i\tau }} , define the Ramanujan G- and g-functions as
2 1 / 4 G n = q − 1 24 ∏ n > 0 ( 1 + q 2 n − 1 ) = η 2 ( τ ) η ( τ 2 ) η ( 2 τ ) , 2 1 / 4 g n = q − 1 24 ∏ n > 0 ( 1 − q 2 n − 1 ) = η ( τ 2 ) η ( τ ) . {\displaystyle {\begin{aligned}2^{1/4}G_{n}&=q^{-{\frac {1}{24}}}\prod _{n>0}(1+q^{2n-1})={\frac {\eta ^{2}(\tau )}{\eta {\big (}{\tfrac {\tau }{2}}{\big )}\eta (2\tau )}},\\2^{1/4}g_{n}&=q^{-{\frac {1}{24}}}\prod _{n>0}(1-q^{2n-1})={\frac {\eta {\big (}{\tfrac {\tau }{2}}{\big )}}{\eta (\tau )}}.\end{aligned}}}The eta quotients make their connection to the first two Weber functions immediately apparent. In the nome, assume τ = − n . {\displaystyle \tau ={\sqrt {-n}}.} Then,
2 1 / 4 G n = f ( q ) = f ( τ ) , 2 1 / 4 g n = f 1 ( q ) = f 1 ( τ ) . {\displaystyle {\begin{aligned}2^{1/4}G_{n}&={\mathfrak {f}}(q)={\mathfrak {f}}(\tau ),\\2^{1/4}g_{n}&={\mathfrak {f}}_{1}(q)={\mathfrak {f}}_{1}(\tau ).\end{aligned}}}Ramanujan found many relations between G n {\displaystyle G_{n}} and g n {\displaystyle g_{n}} which implies similar relations between f ( q ) {\displaystyle {\mathfrak {f}}(q)} and f 1 ( q ) {\displaystyle {\mathfrak {f}}_{1}(q)} . For example, his identity,
( G n 8 − g n 8 ) ( G n g n ) 8 = 1 4 , {\displaystyle (G_{n}^{8}-g_{n}^{8})(G_{n}\,g_{n})^{8}={\tfrac {1}{4}},}leads to
[ f 8 ( q ) − f 1 8 ( q ) ] [ f ( q ) f 1 ( q ) ] 8 = [ 2 ] 8 . {\displaystyle {\big [}{\mathfrak {f}}^{8}(q)-{\mathfrak {f}}_{1}^{8}(q){\big ]}{\big [}{\mathfrak {f}}(q)\,{\mathfrak {f}}_{1}(q){\big ]}^{8}={\big [}{\sqrt {2}}{\big ]}^{8}.}For many values of n, Ramanujan also tabulated G n {\displaystyle G_{n}} for odd n, and g n {\displaystyle g_{n}} for even n. This automatically gives many explicit evaluations of f ( q ) {\displaystyle {\mathfrak {f}}(q)} and f 1 ( q ) {\displaystyle {\mathfrak {f}}_{1}(q)} . For example, using τ = − 5 , − 13 , − 37 {\displaystyle \tau ={\sqrt {-5}},\,{\sqrt {-13}},\,{\sqrt {-37}}} , which are some of the square-free discriminants with class number 2,
G 5 = ( 1 + 5 2 ) 1 / 4 , G 13 = ( 3 + 13 2 ) 1 / 4 , G 37 = ( 6 + 37 ) 1 / 4 , {\displaystyle {\begin{aligned}G_{5}&=\left({\frac {1+{\sqrt {5}}}{2}}\right)^{1/4},\\G_{13}&=\left({\frac {3+{\sqrt {13}}}{2}}\right)^{1/4},\\G_{37}&=\left(6+{\sqrt {37}}\right)^{1/4},\end{aligned}}}and one can easily get f ( τ ) = 2 1 / 4 G n {\displaystyle {\mathfrak {f}}(\tau )=2^{1/4}G_{n}} from these, as well as the more complicated examples found in Ramanujan's Notebooks.
Relation to Jacobi theta functions
The argument of the classical Jacobi theta functions is traditionally the nome q = e π i τ , {\displaystyle q=e^{\pi i\tau },}
ϑ 10 ( 0 ; τ ) = θ 2 ( q ) = ∑ n = − ∞ ∞ q ( n + 1 / 2 ) 2 = 2 η 2 ( 2 τ ) η ( τ ) , ϑ 00 ( 0 ; τ ) = θ 3 ( q ) = ∑ n = − ∞ ∞ q n 2 = η 5 ( τ ) η 2 ( τ 2 ) η 2 ( 2 τ ) = η 2 ( τ + 1 2 ) η ( τ + 1 ) , ϑ 01 ( 0 ; τ ) = θ 4 ( q ) = ∑ n = − ∞ ∞ ( − 1 ) n q n 2 = η 2 ( τ 2 ) η ( τ ) . {\displaystyle {\begin{aligned}\vartheta _{10}(0;\tau )&=\theta _{2}(q)=\sum _{n=-\infty }^{\infty }q^{(n+1/2)^{2}}={\frac {2\eta ^{2}(2\tau )}{\eta (\tau )}},\\[2pt]\vartheta _{00}(0;\tau )&=\theta _{3}(q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\;=\;{\frac {\eta ^{5}(\tau )}{\eta ^{2}\left({\frac {\tau }{2}}\right)\eta ^{2}(2\tau )}}={\frac {\eta ^{2}\left({\frac {\tau +1}{2}}\right)}{\eta (\tau +1)}},\\[3pt]\vartheta _{01}(0;\tau )&=\theta _{4}(q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}={\frac {\eta ^{2}\left({\frac {\tau }{2}}\right)}{\eta (\tau )}}.\end{aligned}}}Dividing them by η ( τ ) {\displaystyle \eta (\tau )} , and also noting that η ( τ ) = e − π i 12 η ( τ + 1 ) {\displaystyle \eta (\tau )=e^{\frac {-\pi i}{\,12}}\eta (\tau +1)} , then they are just squares of the Weber functions f i ( q ) {\displaystyle {\mathfrak {f}}_{i}(q)}
θ 2 ( q ) η ( τ ) = f 2 ( q ) 2 , θ 4 ( q ) η ( τ ) = f 1 ( q ) 2 , θ 3 ( q ) η ( τ ) = f ( q ) 2 , {\displaystyle {\begin{aligned}{\frac {\theta _{2}(q)}{\eta (\tau )}}&={\mathfrak {f}}_{2}(q)^{2},\\[4pt]{\frac {\theta _{4}(q)}{\eta (\tau )}}&={\mathfrak {f}}_{1}(q)^{2},\\[4pt]{\frac {\theta _{3}(q)}{\eta (\tau )}}&={\mathfrak {f}}(q)^{2},\end{aligned}}}with even-subscript theta functions purposely listed first. Using the well-known Jacobi identity with even subscripts on the LHS,
θ 2 ( q ) 4 + θ 4 ( q ) 4 = θ 3 ( q ) 4 ; {\displaystyle \theta _{2}(q)^{4}+\theta _{4}(q)^{4}=\theta _{3}(q)^{4};}therefore,
f 2 ( q ) 8 + f 1 ( q ) 8 = f ( q ) 8 . {\displaystyle {\mathfrak {f}}_{2}(q)^{8}+{\mathfrak {f}}_{1}(q)^{8}={\mathfrak {f}}(q)^{8}.}Relation to j-function
The three roots of the cubic equation
j ( τ ) = ( x − 16 ) 3 x {\displaystyle j(\tau )={\frac {(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 {\displaystyle x_{i}={\mathfrak {f}}(\tau )^{24},-{\mathfrak {f}}_{1}(\tau )^{24},-{\mathfrak {f}}_{2}(\tau )^{24}} . Also, since,
j ( τ ) = 32 ( θ 2 ( q ) 8 + θ 3 ( q ) 8 + θ 4 ( q ) 8 ) 3 ( θ 2 ( q ) θ 3 ( q ) θ 4 ( q ) ) 8 {\displaystyle j(\tau )=32{\frac {{\Big (}\theta _{2}(q)^{8}+\theta _{3}(q)^{8}+\theta _{4}(q)^{8}{\Big )}^{3}}{{\Big (}\theta _{2}(q)\,\theta _{3}(q)\,\theta _{4}(q){\Big )}^{8}}}}and using the definitions of the Weber functions in terms of the Jacobi theta functions, plus the fact that f 2 ( q ) 2 f 1 ( q ) 2 f ( q ) 2 = θ 2 ( q ) η ( τ ) θ 4 ( q ) η ( τ ) θ 3 ( q ) η ( τ ) = 2 {\displaystyle {\mathfrak {f}}_{2}(q)^{2}\,{\mathfrak {f}}_{1}(q)^{2}\,{\mathfrak {f}}(q)^{2}={\frac {\theta _{2}(q)}{\eta (\tau )}}{\frac {\theta _{4}(q)}{\eta (\tau )}}{\frac {\theta _{3}(q)}{\eta (\tau )}}=2} , then
j ( τ ) = ( f ( τ ) 16 + f 1 ( τ ) 16 + f 2 ( τ ) 16 2 ) 3 = ( f ( q ) 16 + f 1 ( q ) 16 + f 2 ( q ) 16 2 ) 3 {\displaystyle j(\tau )=\left({\frac {{\mathfrak {f}}(\tau )^{16}+{\mathfrak {f}}_{1}(\tau )^{16}+{\mathfrak {f}}_{2}(\tau )^{16}}{2}}\right)^{3}=\left({\frac {{\mathfrak {f}}(q)^{16}+{\mathfrak {f}}_{1}(q)^{16}+{\mathfrak {f}}_{2}(q)^{16}}{2}}\right)^{3}}since f i ( τ ) = f i ( q ) {\displaystyle {\mathfrak {f}}_{i}(\tau )={\mathfrak {f}}_{i}(q)} and have the same formulas in terms of the Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )} .
See also
- Duke, William (2005), Continued Fractions and Modular Functions (PDF), Bull. Amer. Math. Soc. 42
- Weber, Heinrich Martin (1981) [1898], Lehrbuch der Algebra (in German), vol. 3 (3rd ed.), New York: AMS Chelsea Publishing, ISBN 978-0-8218-2971-4
- Yui, Noriko; Zagier, Don (1997), "On the singular values of Weber modular functions", Mathematics of Computation, 66 (220): 1645–1662, doi:10.1090/S0025-5718-97-00854-5, MR 1415803
Notes
References
f, f1 and f2 are not modular functions (per the Wikipedia definition), but every modular function is a rational function in f, f1 and f2. Some authors use a non-equivalent definition of "modular functions". /wiki/Modular_form#Modular_functions ↩
https://www.math.ucla.edu/~wdduke/preprints/bams4.pdf Continued Fractions and Modular Functions, W. Duke, pp 22-23 https://www.math.ucla.edu/~wdduke/preprints/bams4.pdf ↩