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 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36fe48a3fdfcfc5fab98a97ef3607e48931fb2b7)
![{\ frac {\ omega _ {1}} {\ omega _ {2}}} \ not \ in {\ mathbb R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a3486701ab423d9d157049ac880cf95b193b7e3)
![\ omega _ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf)
![\ Omega \ subset {\ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1e64f8789deb511516e249ccd3870a7a095c49)
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.
The iron stone row from weight to lattice in is the infinite row of form
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![\ mathbb {C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
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.
These series are absolutely convergent for ; for odd is .
![k \ geq 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd0b1582d6f884e01e27786508ef410fae3de5e4)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![G_ {k} (\ Omega) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/02e2a1bd884ec0abc9a3d755b36637a0c1b72fb4)
Rows of iron stones on the upper half-level
The investigation of the Eisenstein series can be wlog on lattice form with limited, because for a grid with basic always applies:
![{\ mathbb {Z}} + {\ mathbb {Z}} \ tau](https://wikimedia.org/api/rest_v1/media/math/render/svg/25050b48397e8b15ecdea53f02080a5db90c6240)
![\ tau \ in {\ mathbb {H}} = \ {z \ in {\ mathbb {C}} \ mid \ operatorname {Im} z> 0 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d23d035f0561dccbbdcbff1085b264d15ae8626)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![(\ omega _ {1}, \ omega _ {2})](https://wikimedia.org/api/rest_v1/media/math/render/svg/37b78154a0fe04fbe34d83f472c713d70235545f)
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,
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:
![{\ frac {\ omega _ {1}} {\ omega _ {2}}} \ in {\ mathbb {H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d7b35892d7b2af62056fad37dd0a03a3384f5e1)
![{\ displaystyle (1, \ tau), \ \ tau \ in \ mathbb {H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/401947dca9f819119887484c6b064951bbc6ccf9)
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.
The Eisenstein series can therefore be understood as a function on the upper half-plane .
![G_ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d8034a8094aa6549db10b01a1e8f73bb57ac39f)
Eisenstein series are holomorphic in the upper half plane and in the apex ( ).
![{\ displaystyle Im \, z \ to \ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6e471d2ad64607ee4fc5c2e7f6984fbc0b6bb91)
The Eisenstein series is a modular form of weight to the group , that is, for having true
![G_ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d8034a8094aa6549db10b01a1e8f73bb57ac39f)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![{\ text {SL}} _ {2} ({\ mathbb {Z}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d55562a483abd35d51f3699b69792ec8ccae34)
![a, b, c, d \ in {\ mathbb Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/850ea0fcdd7e86e90df9b636cff1829b6e787b7e)
![ad-bc = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/4755f20d723d7bbf809462a07f12f0f5974840bf)
![G_ {k} \! \ Left ({\ frac {a \ tau + b} {c \ tau + d}} \ right) = (c \ tau + d) ^ {k} G_ {k} (\ tau) .](https://wikimedia.org/api/rest_v1/media/math/render/svg/67569ca939eb97119480cd093bb8ab203b816ca8)
For are the polynomials with rational coefficients in and , i. H. , the recursion formula applies :
![k \ geq 8](https://wikimedia.org/api/rest_v1/media/math/render/svg/84ef84ccd6b56db24b692122c769e24ef2ad11ee)
![G_ {4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/149de5cbc9bd78a60484b9c011a9020531255118)
![G_ {6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08981ac4e2b253c7ff0e57783aa846de672afabc)
![G_ {k} \ in {\ mathbb {Q}} [G_ {4}, G_ {6}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ce7c54db0df1ce30df07fab9a37b2b1e349738)
![(n-3) (2n + 1) (2n-1) G _ {{2n}} = 3 \ sum _ {{p = 2}} ^ {{n-2}} (2p-1) (2n-2p -1) G _ {{2p}} G _ {{2n-2p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6edfa10c9d3705173e43f31d5f5eabcb16cb098b)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d928ec15aeef83aade867992ee473933adb6139d)
![7G_ {8} = 3G_ {4} ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a9cea58649eaa2785df5a923f9a024f3a9899c5)
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,
is there
![\, \ sigma _ {k} (n) = \ sum _ {{d | n}} d ^ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68bbcdce7c77979b8f2762fd74b77056ae6fb6d9)
the sum of the -th powers of the divisors of . However, this formula can also be proven elementary (that is, not functionally theoretically).
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
Fourier expansion
The Eisenstein series can be expanded into a Fourier series :
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,
here is the Riemann zeta function . Another common representation is that of the standardized Eisenstein series
![\ zeta (s) = \ sum _ {{n = 1}} ^ {{\ infty}} n ^ {{- s}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d1610c51a022d60c635ed349c446626bd851211)
![{\ 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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c4e422f55610526ca0ed4bee2549673102c77d5)
These are the Bernoulli numbers . This Fourier series has only rational Fourier coefficients.
![B_ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6457760e36cf45e1471e33bcc1536cb4802fb9)
Relation to elliptic functions
It be and . Then the Weierstrasse ℘ function for the lattice satisfies the differential equation
![g_ {2} = 60G_ {4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62a2dcf9e1c8db0b623528d10511d21469c67e82)
![g_ {3} = 140G_ {6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b19924e226a49ed082e86d3779360050bf8d6301)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![(\ wp '(z)) ^ {2} = 4 \ wp (z) ^ {3} -g_ {2} (\ Omega) \ wp (z) -g_ {3} (\ Omega).](https://wikimedia.org/api/rest_v1/media/math/render/svg/acaeebf8f2cacaf55d0f11921c727138220d9382)
Conversely, for every elliptic curve there is over
![y ^ {2} = x ^ {3} + ax + b](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbe6cab1bc2c7f1c99757dc6e5d7a517cf9b4f8)
a grid with and . The elliptic curve is then parameterized by
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![a = 15G_ {4} (\ Omega)](https://wikimedia.org/api/rest_v1/media/math/render/svg/afada07e40c2935199abf153a092b3a5012cc15c)
![b = 35G_ {6} (\ Omega)](https://wikimedia.org/api/rest_v1/media/math/render/svg/97819acfb4411282df3845d27d4de06296ffcf6e)
![(x, y) = (\ wp (z), {\ frac {1} {2}} \ wp '(z))](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3522c2abb1a6d964db8e234b17127b2b6376498)
with . In particular, every elliptic curve is homeomorphic to a torus .
![z \ in {\ mathbb {C}} / \ Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbb0f2c5e177ddb0e213063fdcb1336a5879a81)
![{\ mathbb {C}} / \ Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/471557d890c784380ef9dbfd82f0c600203197b8)
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