Discriminant (modular form)

The discriminant Δ is a holomorphic function on the upper half plane . ${\displaystyle \mathbb {H} =\{z\in \mathbb {C} \mid \mathrm {Im} \,z>0\}}$

It plays an important role in the theory of elliptic functions and modular forms .

definition

For was , ${\displaystyle z\in \mathbb {H} }$ ${\displaystyle \Delta (z):=g_{2}^{3}(z)-27g_{3}^{2}(z)}$

there are and the rows of iron stones to the grid . ${\displaystyle g_{2}(z)=60G_{4}(z)}$ ${\displaystyle g_{3}(z)=140G_{6}(z)}$ ${\displaystyle \mathbb {Z} z+\mathbb {Z} }$

Product development

The discriminant can be developed into an infinite product , the following applies: ${\displaystyle \Delta }$

${\displaystyle \Delta (z)=(2\pi )^{12}e^{2\pi iz}\prod _{n=1}^{\infty }(1-e^{2\pi inz})^{24}}$

From the product representation it follows immediately that in has no zeros. ${\displaystyle \Delta }$ ${\displaystyle \mathbb {H} }$

The discriminant is closely related to Dedekind's η function , it is . ${\displaystyle \Delta }$ ${\displaystyle \Delta (z)=(2\pi )^{12}\eta ^{24}(z)}$

Transformation behavior

The discriminant Δ is a whole modular shape of weight 12, i.e. H. among the substitutions of

${\displaystyle \Gamma :=SL_{2}(\mathbb {Z} )=\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\mid a,b,c,d\in \mathbb {Z} ,ad-bc=1\}}$ applies:

${\displaystyle \Delta \left({\frac {az+b}{cz+d}}\right)=(cz+d)^{12}\Delta (z)}$.

The discriminant Δ has a zero at and is therefore the simplest example of what is known as a cusp shape . ${\displaystyle z=\infty }$

Fourier expansion

The discriminant Δ can be expanded into a Fourier series :

${\displaystyle \Delta (z)=(2\pi )^{12}\sum _{n=1}^{\infty }\tau (n)\,{\mathrm {e} }^{2\pi inz}}$.

The Fourier coefficients are all integers and are called the Ramanujan tau function . This is a multiplicative number theoretic function , i. H.

${\displaystyle \tau (m)\cdot \tau (n)=\tau (m\cdot n)}$for coprime , ${\displaystyle m,n\in \mathbb {N} }$

as proved by Louis Mordell in 1917 . The formula applies more precisely

${\displaystyle \tau (m)\tau (n)=\sum _{d\,|(m,n)}\!\!d^{11}\tau \left({\frac {mn}{d^{2}}}\right)}$.

The following applies to the first values ​​of the tau function : ${\displaystyle \tau (n)}$

${\displaystyle \tau (1)=1}$

${\displaystyle \tau (2)=-24}$

${\displaystyle \tau (3)=252}$.

To date, no “simple” arithmetic definition of the tau function is known. It is also unknown to this day whether the presumption made by Derrick Henry Lehmer

${\displaystyle \tau (m)\neq 0}$is right for everyone . ${\displaystyle m\in \mathbb {N} }$

Ramanujan suggested that for prime numbers the following applies: ${\displaystyle p}$

${\displaystyle |\tau (p)|\leq 2p^{11/2}}$.

This conjecture was proven by Deligne in 1974 .

They fulfill the congruence already discovered by Ramanujan ${\displaystyle \tau (n)}$

${\displaystyle \tau (n)\equiv \sigma _{11}(n)\mod 691}$

With

${\displaystyle \sigma _{11}(n)=\sum _{d\,\mid n}d^{11}}$

literature

• Tom M. Apostol : Modular Functions and Dirichlet Series in Number Theory , Springer, Berlin Heidelberg New York (1990), ISBN 3-540-97127-0
• Eberhard Freitag , Rolf Busam: Funktionentheorie 1 , 4th edition, Springer, Berlin (2006), ISBN 3-540-31764-3
• Max Koecher , Aloys Krieg : Elliptical functions and modular forms , 2nd edition, Springer, Berlin (2007), ISBN 978-3-540-49324-2

Individual evidence

1. ↑ Follow A000594 in OEIS