Let τ be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein seriesG2k(τ) of weight 2k, where k ≥ 2 is an integer, by the following series:[2]
This series absolutely converges to a holomorphic function of τ in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at τ = i∞. It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its SL(2, )-covariance. Explicitly if a, b, c, d ∈ and ad − bc = 1 then
(Proof)
If ad − bc = 1 then
so that
is a bijection 2 → 2, i.e.:
Overall, if ad − bc = 1 then
and G2k is therefore a modular form of weight 2k. Note that it is important to assume that k ≥ 2, otherwise it would be illegitimate to change the order of summation, and the SL(2, )-invariance would not hold. In fact, there are no nontrivial modular forms of weight 2. Nevertheless, an analogue of the holomorphic Eisenstein series can be defined even for k = 1, although it would only be a quasimodular form.
Note that k ≥ 2 is necessary such that the series converges absolutely, whereas k needs to be even otherwise the sum vanishes because the (-m, -n) and (m, n) terms cancel out. For k = 2 the series converges but it is not a modular form.
The article on modular invariants provides expressions for these two functions in terms of theta functions.
Recurrence relation
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Any holomorphic modular form for the modular group[4] can be written as a polynomial in G4 and G6. Specifically, the higher order G2k can be written in terms of G4 and G6 through a recurrence relation. Let dk = (2k + 3)k! G2k + 4, so for example, d0 = 3G4 and d1 = 5G6. Then the dk satisfy the relation
Define q = e2πiτ. (Some older books define q to be the nomeq = eπiτ, but q = e2πiτ is now standard in number theory.) Then the Fourier series of the Eisenstein[5] series is
The third symmetric relation, on the other hand, is a consequence of E8 = E2 4 and a4 − b4 + c4 = 0.
Products of Eisenstein series
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Eisenstein series form the most explicit examples of modular forms for the full modular group SL(2, ). Since the space of modular forms of weight 2k has dimension 1 for 2k = 4, 6, 8, 10, 14, different products of Eisenstein series having those weights have to be equal up to a scalar multiple. In fact, we obtain the identities:[7]
Using the q-expansions of the Eisenstein series given above, they may be restated as identities involving the sums of powers of divisors:
hence
and similarly for the others. The theta function of an eight-dimensional even unimodular lattice Γ is a modular form of weight 4 for the full modular group, which gives the following identities:
Similar techniques involving holomorphic Eisenstein series twisted by a Dirichlet character produce formulas for the number of representations of a positive integer n' as a sum of two, four, or eight squares in terms of the divisors of n.
Using the above recurrence relation, all higher E2k can be expressed as polynomials in E4 and E6. For example:
Many relationships between products of Eisenstein series can be written in an elegant way using Hankel determinants, e.g. Garvan's identity
Srinivasa Ramanujan gave several interesting identities between the first few Eisenstein series involving differentiation.[9] Let
then
These identities, like the identities between the series, yield arithmetical convolution identities involving the sum-of-divisor function. Following Ramanujan, to put these identities in the simplest form it is necessary to extend the domain of σp(n) to include zero, by setting
Then, for example
Other identities of this type, but not directly related to the preceding relations between L, M and N functions, have been proved by Ramanujan and Giuseppe Melfi,[10][11] as for example
Generalizations
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Automorphic forms generalize the idea of modular forms for general Lie groups; and Eisenstein series generalize in a similar fashion.
Defining OK to be the ring of integers of a totally real algebraic number fieldK, one then defines the Hilbert–Blumenthal modular group as PSL(2,OK). One can then associate an Eisenstein series to every cusp of the Hilbert–Blumenthal modular group.
References
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^"Gotthold Eisenstein - Biography". Maths History. Retrieved 2023-09-05.
^Gekeler, Ernst-Ulrich (2011). "PARA-EISENSTEIN SERIES FOR THE MODULAR GROUP GL(2, 𝔽q[T])". Taiwanese Journal of Mathematics. 15 (4): 1463–1475. doi:10.11650/twjm/1500406358. ISSN 1027-5487. S2CID 119499748.
^Obers, N. A.; Pioline, B. (2000-03-07). "Eisenstein Series in String Theory". Classical and Quantum Gravity. 17 (5): 1215–1224. arXiv:hep-th/9910115. Bibcode:2000CQGra..17.1215O. doi:10.1088/0264-9381/17/5/330. ISSN 0264-9381. S2CID 250864942.
^Mertens, Michael H.; Rolen, Larry (2015). "Lacunary recurrences for Eisenstein series". Research in Number Theory. 1. arXiv:1504.00356. doi:10.1007/s40993-015-0010-x. ISSN 2363-9555.
^Karel, Martin L. (1974). "Fourier Coefficients of Certain Eisenstein Series". Annals of Mathematics. 99 (1): 176–202. doi:10.2307/1971017. ISSN 0003-486X. JSTOR 1971017.
^"How to prove this series identity involving Eisenstein series?". Mathematics Stack Exchange. Retrieved 2023-09-05.
^Dickson, Martin; Neururer, Michael (2018). "Products of Eisenstein series and Fourier expansions of modular forms at cusps". Journal of Number Theory. 188: 137–164. arXiv:1603.00774. doi:10.1016/j.jnt.2017.12.013. S2CID 119614418.
^Milne, Steven C. (2000). "Hankel Determinants of Eisenstein Series". arXiv:math/0009130v3. The paper uses a non-equivalent definition of , but this has been accounted for in this article.
^Bhuvan, E. N.; Vasuki, K. R. (2019-06-24). "On a Ramanujan's Eisenstein series identity of level fifteen". Proceedings - Mathematical Sciences. 129 (4): 57. doi:10.1007/s12044-019-0498-4. ISSN 0973-7685. S2CID 255485301.
^Ramanujan, Srinivasa (1962). "On certain arithmetical functions". Collected Papers. New York, NY: Chelsea. pp. 136–162.
^Melfi, Giuseppe (1998). "On some modular identities". Number Theory, Diophantine, Computational and Algebraic Aspects: Proceedings of the International Conference held in Eger, Hungary. Walter de Grutyer & Co. pp. 371–382.
Further reading
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Akhiezer, Naum Illyich (1970). Elements of the Theory of Elliptic Functions (in Russian). Moscow.{{cite book}}: CS1 maint: location missing publisher (link) Translated into English as Elements of the Theory of Elliptic Functions. AMS Translations of Mathematical Monographs 79. Providence, RI: American Mathematical Society. 1990. ISBN 0-8218-4532-2.
Apostol, Tom M. (1990). Modular Functions and Dirichlet Series in Number Theory (2nd ed.). New York, NY: Springer. ISBN 0-387-97127-0.
Chan, Heng Huat; Ong, Yau Lin (1999). "On Eisenstein Series" (PDF). Proc. Amer. Math. Soc. 127 (6): 1735–1744. doi:10.1090/S0002-9939-99-04832-7.
Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics 7 (transl. ed.). New York & Heidelberg: Springer-Verlag. ISBN 9780387900407.