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Eisenstein series are modular forms; their Fourier coefficients

Statement

For every even $k \geq 4$ : (a) $E_{k}$ is a modular form of weight $k$ for $P S L_{2} (Z)$ ; (b) writing $B_{k}$ for the $k$ -th Bernoulli number, $E_{k} (τ) = 1 - \frac{2 k}{B_{k}} \sum_{n \geq 1} σ_{k - 1} (n) q^{n}$ , $q = e^{2 π i τ}$ , with $σ_{k - 1} (n) = \sum_{d ∣ n} d^{k - 1}$ ; (c) $E_{k} (\infty) = 1$ , and $G_{k} = 2 ζ (k) E_{k}$ ; (d) in particular $E_{4} = 1 + 240 q + 2160 q^{2} + O (q^{3})$ and $E_{6} = 1 - 504 q - 16632 q^{2} + O (q^{3})$ .

Facts & Assumptions

Given: Even $k \geq 4$ , $G_{k} (τ) = \sum_{(m, n)}^{'} (m τ + n)^{- k}$ and $E_{k} = G_{k} / (2 ζ (k))$ (The level-one Eisenstein series E_k and the weight-two series E_2, The Riemann zeta function on the half-plane $Re s > 1$ , The divisor power sums $σ_{k}$ ).

[F1]

The family $((m τ + n)^{- k})$ is absolutely summable with locally uniform convergence on $H$ , and the action is by $ℑ (γ τ) = ℑ τ / ∣ c τ + d ∣^{2}$ with $γ = (\begin{matrix} a & b \\ c & d \end{matrix})$ (Absolute convergence and holomorphy of the lattice Eisenstein sums, The modular group and its action on the upper half-plane).

[F2]

Lipschitz: $\sum_{n \in Z} (z + n)^{- k} = \frac{(- 2 π i)^{k}}{(k - 1)!} \sum_{r \geq 1} r^{k - 1} e^{2 π i r z}$ for $Im z > 0$ , absolutely convergent (The Lipschitz formula for the reciprocal-power sums).

[F3]

$M_{k}$ is defined by the weight- $k$ transformation law and holomorphy at the cusp, with $E_{k} = 1 + O (q)$ (Level-one modular forms and cusp forms, The level-one Eisenstein series E_k and the weight-two series E_2); $2 ζ (k) > 0$ and $(m, n) \mapsto (m a + n c, m b + n d)$ is a bijection of $Z^{2}$ for $γ \in S L_{2} (Z)$ (The Riemann zeta function on the half-plane $Re s > 1$ , Fubini for double series: if $\sum_{i} \sum_{j} ∣ a_{i j} ∣$ converges then both iterated sums and the sum along every bijection $N \to N \times N$ converge to one and the same value).

[F4]

The even zeta values follow locally without a choice assumption. For $∣ z ∣ < 1$ , the cotangent expansion (The Mittag-Leffler expansion of pi cotangent) gives $π z \cot (π z) = 1 - 2 \sum_{m \geq 1} ζ (2 m) z^{2 m}$ : expand $2 z^{2} / (z^{2} - n^{2})$ geometrically and interchange the sums, whose absolute total on $∣ z ∣ \leq r < 1$ is bounded by $2 r^{2} (1 - r^{2})^{- 1} \sum_{n \geq 1} n^{- 2}$ . Put $t = 2 π i z$ . The exponential definitions of sine and cosine give $π z \cot (π z) = t / (e^{t} - 1) + t / 2$ . Comparing the coefficient of $z^{2 m}$ in the Bernoulli generating series yields $ζ (2 m) = (- 1)^{m + 1} B_{2 m} (2 π)^{2 m} / (2 (2 m)!)$ (The Bernoulli numbers are defined by the generating series $t / (e^{t} - 1)$ , Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential, $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential, Fubini for double series: if $\sum_{i} \sum_{j} ∣ a_{i j} ∣$ converges then both iterated sums and the sum along every bijection $N \to N \times N$ converge to one and the same value, Ratio test: $lim sup ∣ a_{k + 1} / a_{k} ∣ < 1$ gives absolute convergence and hence convergence, and $lim inf ∣ a_{k + 1} / a_{k} ∣ > 1$ gives divergence).

Proof

1.1F1F3givenalgebra

Let $γ = (\begin{matrix} a & b \\ c & d \end{matrix}) \in S L_{2} (Z)$ . Writing $m γ τ + n = \frac{(m a + n c) τ + (m b + n d)}{c τ + d}$ and using that $(m, n) \mapsto (m a + n c, m b + n d)$ is a bijection [F3], the absolute summability [F1] permits reindexing, so $G_{k} (γ τ) = (c τ + d)^{k} G_{k} (τ)$ . Hence $E_{k} (γ τ) = (c τ + d)^{k} E_{k} (τ)$ . By [F3], $E_{k} = 1 + O (q)$ as $Im τ \to \infty$ , so in particular $E_{k}$ is bounded near the cusp, and its periodic function $F$ extends holomorphically to $q = 0$ by the q-expansion principle; therefore $E_{k} \in M_{k}$ and (a) holds.

1.2F2F3givenalgebra

Split the absolutely summable sum defining $G_{k}$ into $m = 0$ and $m \neq 0$ . The $m = 0$ part is $\sum_{n \neq 0} n^{- k} = 2 ζ (k)$ because $k$ is even. For $m \neq 0$ pair $m$ with $- m$ and $n$ with $- n$ ; by [F3] this reindexing preserves the sum, and each remaining term with $m \geq 1$ is handled by [F2] applied to $z = m τ$ (whose imaginary part is positive): $\sum_{n \in Z} (m τ + n)^{- k} = \frac{(- 2 π i)^{k}}{(k - 1)!} \sum_{r \geq 1} r^{k - 1} q^{m r}$ . Therefore $G_{k} = 2 ζ (k) + 2 \frac{(- 2 π i)^{k}}{(k - 1)!} \sum_{m \geq 1} \sum_{r \geq 1} r^{k - 1} q^{m r} = 2 ζ (k) + 2 \frac{(- 2 π i)^{k}}{(k - 1)!} \sum_{n \geq 1} σ_{k - 1} (n) q^{n}$ , the last regrouping being the absolutely summable family $(r^{k - 1} q^{m r})_{m, r \geq 1}$ grouped by $n = m r$ [F3]: for $∣ q ∣ \leq ρ < 1$ , its absolute sum is at most $(1 - ρ)^{- 1} \sum_{r \geq 1} r^{k - 1} ρ^{r} < \infty$ by the ratio test (Ratio test: $lim sup ∣ a_{k + 1} / a_{k} ∣ < 1$ gives absolute convergence and hence convergence, and $lim inf ∣ a_{k + 1} / a_{k} ∣ > 1$ gives divergence). Dividing by $2 ζ (k)$ gives $E_{k} = 1 + \frac{(- 2 π i)^{k}}{(k - 1)! ζ (k)} \sum_{n \geq 1} σ_{k - 1} (n) q^{n}$ .

2.1F4step 1.2givenalgebra∎

For even $k$ write $k = 2 m$ . Then $(- 2 π i)^{k} = 2^{k} π^{k} (- i)^{2 m} = (- 1)^{m} 2^{k} π^{k}$ , and by [F4] $ζ (2 m) = (- 1)^{m + 1} \frac{B_{k} (2 π)^{k}}{2 k!}$ , so $(k - 1)! ζ (k) = (- 1)^{m + 1} \frac{B_{k} (2 π)^{k}}{2 k}$ and the coefficient of $\sum σ_{k - 1} (n) q^{n}$ in 1.2 is $\frac{(- 1)^{m} 2^{k} π^{k} \cdot 2 k}{(- 1)^{m + 1} B_{k} (2 π)^{k}} = - \frac{2 k}{B_{k}}$ , proving (b). The constant term gives $E_{k} (\infty) = 1$ and $G_{k} = 2 ζ (k) E_{k}$ , which is (c) (see [F3]). For (d): $B_{4} = - 1 / 30$ and $B_{6} = 1 / 42$ , so $- \frac{2 \cdot 4}{B_{4}} = 240$ and $- \frac{2 \cdot 6}{B_{6}} = - 504$ ; with $σ_{3} (1) = 1$ , $σ_{3} (2) = 1 + 8 = 9$ and $σ_{5} (1) = 1$ , $σ_{5} (2) = 1 + 32 = 33$ this gives $E_{4} = 1 + 240 q + 2160 q^{2} + O (q^{3})$ and $E_{6} = 1 - 504 q - 16632 q^{2} + O (q^{3})$ , as claimed. The special values used here are the local coefficient computation in [F4].

Depends on

Used by

Integrality of the Fourier coefficients of the j-invariant Corollary
The dimension of the space of level-one modular forms Corollary
The zeros of E4 and E6 at the elliptic points Corollary
The modular discriminant and the j-invariant Definition
The first Fourier coefficients of E4, E6, Delta and j Example
The discriminant is a nonvanishing cusp form of weight 12 Lemma
The j-invariant of the Legendre normal form Lemma
The graded ring of level-one modular forms Theorem
The level-one valence formula Theorem

Dependency tree · two levels

112 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

D. Zagier, Elliptic Modular Forms and Their Applications, in The 1-2-3 of Modular Forms (Universitext, Springer, 2008) (standard reference, not scraped)
J. S. Milne, Modular Functions and Modular Forms (v1.31, 2017) (standard reference, not scraped)
C. T. McMullen, Advanced Complex Analysis, Math 213a course notes (Harvard, 2010) (standard reference, not scraped)