Eisenstein range

Eisensteinreihen (after the German mathematician Gotthold Eisenstein ) are different series from the theory of modular forms or automorphic forms .

Holomorphic Eisenstein series

Rows of iron stones on the space of the bars

Let be two complex numbers with . The grid created by and is ${\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} \setminus \{0\}}$ ${\displaystyle {\frac {\omega _{1}}{\omega _{2}}}\not \in \mathbb {R} }$ ${\displaystyle \omega _{1}}$ ${\displaystyle \omega _{2}}$ ${\displaystyle \Omega \subset \mathbb {C} }$

${\displaystyle \Omega :=\left\{\omega =a\omega _{1}+b\omega _{2}:a,b\in \mathbb {Z} \right\}}$.

The iron stone row from weight to lattice in is the infinite row of form ${\displaystyle k}$ ${\displaystyle \Omega }$ ${\displaystyle \mathbb {C} }$

${\displaystyle G_{k}(\Omega ):=\sum _{0\not =\omega \in \Omega }\omega ^{-k}}$.

These series are absolutely convergent for ; for odd is . ${\displaystyle k\geq 3}$ ${\displaystyle k}$ ${\displaystyle G_{k}(\Omega )=0}$

Rows of iron stones on the upper half-level

${\displaystyle G_{4}}$

${\displaystyle G_{6}}$

${\displaystyle G_{8}}$

${\displaystyle G_{10}}$

${\displaystyle G_{12}}$

${\displaystyle G_{14}}$

The investigation of the Eisenstein series can be wlog on lattice form with limited, because for a grid with basic always applies: ${\displaystyle \mathbb {Z} +\mathbb {Z} \tau }$ ${\displaystyle \tau \in \mathbb {H} =\{z\in \mathbb {C} \mid \operatorname {Im} z>0\}}$ ${\displaystyle \Omega }$ ${\displaystyle (\omega _{1},\omega _{2})}$

${\displaystyle G_{k}(\Omega )=G_{k}(\omega _{1},\omega _{2})=\omega _{2}^{-k}G_{k}\left({\frac {\omega _{1}}{\omega _{2}}},1\right)}$,

and since the basis can be chosen to hold, one can compute the Eisenstein rows of any lattice once one knows them for those with basis . The latter is also abbreviated to: ${\displaystyle {\frac {\omega _{1}}{\omega _{2}}}\in \mathbb {H} }$ ${\displaystyle (1,\tau ),\ \tau \in \mathbb {H} }$

${\displaystyle G_{k}(\tau ):=G_{k}(\tau ,1)=\sum _{(0,0)\not =(m,n)\in \mathbb {Z} \times \mathbb {Z} }{\frac {1}{(m\tau +n)^{k}}}}$.

The Eisenstein series can therefore be understood as a function on the upper half-plane . ${\displaystyle G_{k}}$

Eisenstein series are holomorphic in the upper half plane and in the apex ( ). ${\displaystyle Im\,z\to \infty }$

The Eisenstein series is a modular form of weight to the group , that is, for having true ${\displaystyle G_{k}}$ ${\displaystyle k}$ ${\displaystyle {\text{SL}}_{2}(\mathbb {Z} )}$ ${\displaystyle a,b,c,d\in \mathbb {Z} }$ ${\displaystyle ad-bc=1}$

${\displaystyle G_{k}\!\left({\frac {a\tau +b}{c\tau +d}}\right)=(c\tau +d)^{k}G_{k}(\tau ).}$

For are the polynomials with rational coefficients in and , i. H. , the recursion formula applies : ${\displaystyle k\geq 8}$ ${\displaystyle G_{k}}$ ${\displaystyle G_{4}}$ ${\displaystyle G_{6}}$ ${\displaystyle G_{k}\in \mathbb {Q} [G_{4},G_{6}]}$

${\displaystyle (n-3)(2n+1)(2n-1)G_{2n}=3\sum _{p=2}^{n-2}(2p-1)(2n-2p-1)G_{2p}G_{2n-2p}}$

Especially for results from this Fourier developments and by a coefficient comparison (see below) the remarkable number theoretic Hurwitz identity (by Adolf Hurwitz ): ${\displaystyle n=4}$ ${\displaystyle 7G_{8}=3G_{4}^{2}}$

${\displaystyle \sigma _{7}(m)=\sigma _{3}(m)+120\sum _{r,s\in \mathbb {N} ,r+s=m}\sigma _{3}(r)\sigma _{3}(s)}$,

is there

${\displaystyle \,\sigma _{k}(n)=\sum _{d|n}d^{k}}$

the sum of the -th powers of the divisors of . However, this formula can also be proven elementary (that is, not functionally theoretically). ${\displaystyle k}$ ${\displaystyle n}$

Fourier expansion

The Eisenstein series can be expanded into a Fourier series :

${\displaystyle G_{k}(\tau )=2\zeta (k)+2{\frac {(2\pi i)^{k}}{(k-1)!}}\sum _{m=1}^{\infty }\sigma _{k-1}(m)e^{2\pi im\tau }}$,

here is the Riemann zeta function . Another common representation is that of the standardized Eisenstein series ${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }n^{-s}}$

${\displaystyle G_{k}^{*}(\tau )={\frac {1}{2\zeta (k)}}G_{k}(\tau )=1-{\frac {2k}{B_{k}}}\sum _{m=1}^{\infty }\sigma _{k-1}(m)e^{2\pi im\tau }.}$

These are the Bernoulli numbers . This Fourier series has only rational Fourier coefficients. ${\displaystyle B_{k}}$

Relation to elliptic functions

It be and . Then the Weierstrasse ℘ function for the lattice satisfies the differential equation ${\displaystyle g_{2}=60G_{4}}$ ${\displaystyle g_{3}=140G_{6}}$ ${\displaystyle \Omega }$

${\displaystyle (\wp '(z))^{2}=4\wp (z)^{3}-g_{2}(\Omega )\wp (z)-g_{3}(\Omega ).}$

Conversely, for every elliptic curve there is over ${\displaystyle \mathbb {C} }$

${\displaystyle y^{2}=x^{3}+ax+b}$

a grid with and . The elliptic curve is then parameterized by ${\displaystyle \Omega }$ ${\displaystyle a=15G_{4}(\Omega )}$ ${\displaystyle b=35G_{6}(\Omega )}$

${\displaystyle (x,y)=(\wp (z),{\frac {1}{2}}\wp '(z))}$

with . In particular, every elliptic curve is homeomorphic to a torus . ${\displaystyle z\in \mathbb {C} /\Omega }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} /\Omega }$

literature

• Eberhard Freitag & Rolf Busam: Function Theory 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