Eisenstein series ($E_k$)

An Eisenstein series is a modular form constructed by summing a weight factor over a parabolic coset space. For even $k>2$, the level-one holomorphic Eisenstein series is a scalar multiple of ${\sum }_{(m,n)\ne (0,0)}(m\tau +n{)}^{-k}$.

For a period lattice $\Lambda \subset \mathbb{C}$ and an integer $k>2$, this sum converges absolutely. Pairing $\omega$ and $-\omega$ makes it zero when $k$ is odd. For even weights it supplies the Laurent coefficients of the Weierstrass elliptic function. Scaling the period lattice gives $G_k(a\Lambda )=a^{-k}{G}_k(\Lambda )$.

Equal sums give equal Klein j-invariants, hence the complex lattices are homothetic. If both sums are nonzero, the scaling factor obeys $a^4=a^6=1$ and is $\pm 1$. If only $G_6$ vanishes, the lattice is homothetic to the square lattice and the possible fourth roots of unity already preserve it. If only $G_4$ vanishes, it is homothetic to the hexagonal lattice and the possible sixth roots of unity preserve it. The two sums cannot vanish together because the elliptic discriminant is nonzero.

There are no weight-four cusp forms at level two. A nonzero form would have a weight-twelve coset norm of a modular form with modular cusp order at least two, contradicting the simple modular cusp order and interior nonvanishing of the modular discriminant. The two independent displayed Eisenstein series span the weight-four space, since its cusp-constant map embeds it in a two-dimensional space.

For $k>2$ and a Dirichlet character modulo $N$ with $\chi (-1)=(-1{)}^k$, define $G_k(\chi ,z)={\sum }_{m,n\in \mathbb{Z},(n,N)=1}\chi (n)(mz+n{)}^{-k}$. This series converges absolutely and locally uniformly on the complex upper half-plane and has period $N$.

With $S_{\chi }(r)$ the finite Fourier transform in Gauss sum of a Dirichlet character, the constant coefficient of $G_k(\chi ,z)$ in $e^{2\pi iz/N}$ is $2L(\chi ,k)$, and its positive coefficients areFor a primitive Dirichlet character this simplifies to $2(-2\pi i{)}^{k}g(\chi )((k-1)!N^k{)}^{-1}{\sum }_{r\mid n}\overline{\chi }(r)r^{k-1}$. Without primitivity the simplified formula is generally false.

For even $k\ge 4$, the normalized level-one Eisenstein series has expansion

The identity $E_8=E_4^2$ gives

If $N+1=\dim M_k(S{L}_2(\mathbb{Z}))$, there is a basis $f_0,\dots ,f_N$ with integral Fourier coefficients andStart from the integral forms ${\Delta }^i{E}_4^a{E}_6^b$ and use integer elimination on their unitriangular leading coefficients.

If the first nonconstant coefficient of $E_k$ has reduced denominator divisible by a prime $p$, the integral echelon basis produces an integral cusp form whose Fourier coefficients are congruent modulo $p$ to ${\sigma }_{k-1}(n)$.

The weight-sixteen cusp form space is spanned by $E_4\Delta$. Its leading coefficient and the Fourier expansion of a normalized Eisenstein series give $3617E_{16}=3617E_4^4-3456000E_4\Delta$. The coefficient multiplier $-3456000$ is congruent to $16320$ modulo $3617$, and is invertible there. Coefficient comparison gives the displayed modular congruence. Integrality of $E_4\Delta$ follows from the modular discriminant product.

For $k>2$ and $(x,y)\in (\mathbb{Z}/N\mathbb{Z}{)}^2$,is a modular form of weight $k$ for the principal congruence subgroup $\Gamma (N)$. Right multiplication of $(x,y)$ by $\gamma \in S{L}_2(\mathbb{Z})$ describes its slash transformation.

For the full modular group, a cusp form is orthogonal under the Petersson inner product to every holomorphic Eisenstein series of the same weight. Unfolding reduces the integral to the constant Fourier coefficient of the cusp form, which is zero.

The seriesis quasimodular: $E_2(-1/\tau )={\tau }^2{E}_2(\tau )+6\tau /(\pi i)$.

The logarithmic derivative of the modular discriminant gives $E_2={\Delta }^{\prime }/(2\pi i\Delta )$. Differentiating its weight-twelve transformation yields the anomalous term $6c(c\tau +d)/(\pi i)$. Subtracting the displayed inverse-height term cancels that anomaly. The result transforms with weight two under the modular group but is not holomorphic, so it is not a holomorphic modular form.

The finite Laurent coefficient $C(z)$ in the Kronecker limit formula is invariant under the modular group. Its Wirtinger derivative satisfies ${\partial }_{z}C=-\pi i(E_2(z)-3/(\pi y))/12$, by differentiating the convergent product in that formula. Differentiating $C(\gamma z)=C(z)$ gives weight-two covariance for $E_2^{\ast }$. Since $\Im (\gamma z)=y/|cz+d{|}^2$, substitution gives the displayed transformation law. This proof needs no prior modularity assertion for the eta product.

For a matrix in the Gamma 0 congruence subgroup, argument dilation conjugates its action to an integral determinant-one matrix for each divisor $M$ of $N$. The anomaly of $E_2(M\tau )$ is $6c(c\tau +d)/(M\pi i)$, so the displayed linear condition cancels it. Necessity follows by testing the matrix with rows $(1,0)$ and $(N,1)$. The combinations $E_2(\tau )-M{E}_2(M\tau )$, with $M>1$ dividing $N$, span this space.

In the displayed iterated order, with the $n$ sum evaluated first, $G_2(z)=({\pi }^2/3)E_2(z)$. The cosecant partial-fraction identity ${\sum }_n(w+n{)}^{-2}={\pi }^2{\csc }^2(\pi w)$ gives the positive-$m$ contribution $-4{\pi }^2{\sum }_{r\ge 1}r{q}^{mr}$; negative $m$ gives the same contribution. Including the $m=0$ term yields ${\pi }^2/3-8{\pi }^2{\sum }_{n\ge 1}{\sigma }_1(n)q^n$. The full lattice series does not have absolute convergence, so this order must not be replaced by arbitrary rearrangement.

For $\mathrm{Re}s>1$,is an absolutely convergent modular-invariant function.

This modular-invariant function, initially defined for $\Re s>1$, has Fourier expansionwhere $\Xi (u)={\pi }^{-u/2}\Gamma (u/2)\zeta (u)$ is the completed Riemann zeta function, without the entire-xi polynomial factor. Poisson summation and the integral representation of the modified Bessel function of the second kind prove the expansion. It gives meromorphic continuation, $\mathcal{E}(z,s)=\mathcal{E}(z,1-s)$, and simple poles at zero and one with residues $-1/2,1/2$. Apparent poles at $s=1/2$ cancel between the two constant terms.

This normalization of the first Kronecker limit formula follows from the Fourier expansion of the completed nonholomorphic Eisenstein series. At $s=1$, $K_{1/2}(t)=\sqrt{\pi /(2t)}{e}^{-t}$, so the nonconstant terms sum to $-2\log |{\prod }_{m\ge 1}(1-q^m)|$. The constant terms supply $\pi y/6-(\log y)/2+(\gamma -\log (4\pi ))/2$. Combining them gives the Dedekind eta function expression, with $\gamma$ the Euler--Mascheroni constant.

For $\mathrm{Re}s>1$ this Eisenstein series also equals ${\sum }_{\gamma \in {\Gamma }_{\infty }\backslash S{L}_2(\mathbb{Z})}(\mathrm{Im}\gamma \tau {)}^s$, where ${\Gamma }_{\infty }$ consists of translations and their negatives. It is modular invariant. The nonholomorphic Eisenstein series summed over all nonzero integer pairs instead equals $2\zeta (2s)E(\tau ,s)$: separate each pair into a positive integer times a primitive pair. This normalization matters in Rankin–Selberg unfolding.

For the series over all nonzero integer pairs, the constant Fourier coefficient is $2\zeta (2s)y^s+2\sqrt{\pi }\Gamma (s-1/2)\zeta (2s-1)y^{1-s}/\Gamma (s)$. Integrating over one real period unfolds the translated intervals and then uses the Gaussian integral. Multiplication by ${\pi }^{-s}\Gamma (s)$ expresses it as $2\Lambda (2s)y^s+2\Lambda (2s-1)y^{1-s}$ in terms of the completed Riemann zeta function.

For the normalization $(T_{p}f)(\tau )=p^{-1}(f(p\tau )+{\sum }_{b\mathrm{mod}p}f((\tau +b)/p))$, one has

For even $k$ and $\mathrm{Re}s>(2-k)/2$, defineIt is generally nonholomorphic and transforms with weight $k$.

For fixed $\tau$, Mellin transformation of a weighted Gaussian theta series expresses $G_k(\tau ,s)$ as a gamma factor times a Mellin integral. Splitting at one and applying Poisson summation to the small-time part continues it to all $s\in \mathbb{C}$; for positive even $k$, the reciprocal gamma factor cancels the apparent poles.

The absolute value of the summand indexed by a primitive bottom row $(c,d)$ isThe exponent exceeds two, so comparison with the lattice sum over $(c,d)\in {\mathbb{Z}}^2\setminus \{0\}$ proves absolute and locally uniform convergence.

If $f$ is a weight-$k$ modular form, thenis modular invariant. The factors transform by $j(\gamma ,\tau {)}^k$, ${\overline{j(\gamma ,\tau )}}^k$, and $|j(\gamma ,\tau ){|}^{-2k}$.