Eisenstein Series and Bernoulli Numbers¶

We introduce generalized Bernoulli numbers attached to Dirichlet characters and give an algorithm to enumerate the Eisenstein series in $M_k(N,\eps)$ .

The Eisenstein Subspace¶

Let $M_k(\Gamma_1(N))$ be the space of modular forms of weight $k$ for $\Gamma_1(N)$ , and let $\T$ be the Hecke algebra acting on $M_k(\Gamma_1(N))$ , which is the subring of $\End(M_k(\Gamma_1(N)))$ generated by all Hecke operators. Then there is a $\T$ -module decomposition

$M_k(\Gamma_1(N)) = \Eis_k(\Gamma_1(N)) \oplus S_k(\Gamma_1(N)),$

where $S_k(\Gamma_1(N))$ is the subspace of modular forms that vanish at all cusps and $\Eis_k(\Gamma_1(N))$ is the Eisenstein subspace, which is uniquely determined by this decomposition. The above decomposition induces a decomposition of $M_k(\Gamma_0(N))$ and of $M_k(N,\eps)$ , for any Dirichlet character $\eps$ of modulus $N$ .

Generalized Bernoulli Numbers¶

Suppose $\eps$ is a Dirichlet character of modulus $N$ over $\C$ . Leopoldt [Leo58] defined generalized Bernoulli numbers attached to $\eps$ .

Definition 5.1

We define the generalized Bernoulli numbers $B_{k,\eps}$ attached to $\eps$ by the following identity of infinite series:

$\sum_{a=1}^{N} \frac{\eps(a) \cdot x \cdot e^{ax}}{e^{Nx}-1} \,\, = \,\, \sum_{k=0}^{\infty} B_{k,\eps} \cdot \frac{x^k}{k!}.$

If $\eps$ is the trivial character of modulus $1$ and $B_k$ are as in Section Examples of Modular Forms of Level 1, then $B_{k,\eps} = B_k$ , except when $k=1$ , in which case $B_{1,\eps} = -B_1 = 1/2$ (see Exercise 5.2).

Algebraically Computing Generalized Bernoulli Numbers¶

Let $\Q(\eps)$ denote the field generated by the image of the character $\eps$ ; thus $\Q(\eps)$ is the cyclotomic extension $\Q(\zeta_n)$ , where $n$ is the order of $\eps$ .

Algorithm 5.2

Given an integer $k\geq 0$ and any Dirichlet character $\eps$ with modulus $N$ , this algorithm computes the generalized Bernoulli numbers $B_{j,\eps}$ , for $j\leq k$ .

Compute $g \set x/(e^{Nx}-1) \in \Q[[x]]$ to precision $O(x^{k+1})$ by computing $e^{Nx}-1 = \sum_{n\geq 1} N^n x^n/n!$ to precision $O(x^{k+2})$ and computing the inverse $1/(e^{Nx}-1)$ , then multiplying by $x$ .
For each $a=1,\ldots, N$ , compute $f_a \set g \cdot e^{ax}\in\Q[[x]]$ , to precision $O(x^{k+1})$ . This requires computing $e^{ax}=\sum_{n\geq 0} a^n x^n/n!$ to precision $O(x^{k+1})$ . (Omit computation of $e^{Nx}$ if $N>1$ since then $\eps(N)=0$ .)
Then for $j\leq k$ , we have

$B_{j,\eps} \set j!\cdot \sum_{a=1}^{N} \eps(a) \cdot c_j(f_a),$
where $c_j(f_a)$ is the coefficient of $x^j$ in $f_a$ .

Note that in steps (1) and (2) we compute the power series doing arithmetic only in $\Q[[x]]$ , not in $\Q(\eps)[[x]]$ , which could be much less efficient if $\eps$ has large order. In step (1) if $k$ is huge, we could compute the inverse $1/(e^{Nx}-1)$ using asymptotically fast arithmetic and Newton iteration.

Example 5.3

The nontrivial character $\eps$ with modulus $4$ has order $2$ and takes values in $\Q$ . The Bernoulli numbers $B_{k,\eps}$ for $k$ even are all $0$ and for $k$ odd they are

$B_{1,\eps} &= -1/2,\\ B_{3,\eps} &= 3/2,\\ B_{5,\eps} &= -25/2,\\ B_{7,\eps} &= 427/2,\\ B_{9,\eps} &= -12465/2,\\ B_{11,\eps} &= 555731/2,\\ B_{13,\eps} &= -35135945/2,\\ B_{15,\eps} &= 2990414715/2,\\ B_{17,\eps} &= -329655706465/2,\\ B_{19,\eps} &= 45692713833379/2.$

Example 5.4

The generalized Bernoulli numbers need not be in $\Q$ . Suppose $\eps$ is the mod $5$ character such that $\eps(2) = i = \sqrt{-1}$ . Then $B_{k,\eps}=0$ for $k$ even and

$B_{1,\eps} &= \frac{-i - 3}{5},\\ B_{3,\eps} &= \frac{6i + 12}{5},\\ B_{5,\eps} &= \frac{-86i - 148}{5},\\ B_{7,\eps} &= \frac{2366i + 3892}{5},\\ B_{9,\eps} &= \frac{-108846i - 176868}{5},\\ B_{11,\eps} &= \frac{7599526i + 12309572}{5},\\ B_{13,\eps} &= \frac{-751182406i - 1215768788}{5},\\ B_{15,\eps} &= \frac{99909993486i + 161668772052}{5},\\ B_{17,\eps} &= \frac{-17209733596766i - 27846408467908}{5}.\\$

Example 5.5

We use Sage to compute some of the above generalized Bernoulli numbers. First we define the character and verify that $\eps(2)=i$ (note that in Sage zeta4 is $\sqrt{-1}$ ).

sage: G = DirichletGroup(5)
sage: e = G.0
sage: e(2)
zeta4

We compute the Bernoulli number $B_{1,\eps}$ .

sage: e.bernoulli(1)
-1/5*zeta4 - 3/5

We compute $B_{9,\eps}$ .

sage: e.bernoulli(9)
-108846/5*zeta4 - 176868/5

Proposition 5.6

If $\eps(-1) \neq (-1)^k$ and $k\geq 2$ , then $B_{k,\eps}=0$ .

Proof

See Exercise 5.3.

Computing Generalized Bernoulli Numbers Analytically¶

This section, which was written jointly with Kevin McGown, is about a way to compute generalized Bernoulli numbers, which is similar to the algorithm in Section Fast Computation of Bernoulli Numbers.

Let $\chi$ be a primitive Dirichlet character modulo its conductor $f$ . Note from the definition of Bernoulli numbers that if $\sigma\in\Gal(\Qbar/\Q)$ , then

(1) $\sigma(B_{n,\chi}) = B_{n,\sigma(\chi)}.$

For any character $\chi$ , we define the Gauss sum $\tau(\chi)$ as

$\tau(\chi)=\sum_{r=1}^{f-1} \chi(r)\,\zeta^r \,,$

where $\zeta=\exp(2\pi i/f)$ is the principal $f^{th}$ root of unity. The Dirichlet $L$ -function for $\chi$ for $\Re(s)>1$ is

$L(s,\chi)=\sum_{n=1}^\infty \chi(n)\,n^{-s} \,.$

In the right half plane $\{s\in\C\mid \Re(s)>1\}$ this function is analytic, and because $\chi$ is multiplicative, we have the Euler product representation

(2) $L(s,\chi)=\prod_{p \text{ prime}} \left(1-\chi(p) p^{-s}\right)^{-1} \,.$

We note (but will not use) that through analytic continuation $L(s,\chi)$ can be extended to a meromorphic function on the entire complex plane.

If $\chi$ is a nonprincipal primitive Dirichlet character of conductor $f$ such that $\chi(-1)=(-1)^n$ , then (see, e.g., [Wan82])

$L(n,\chi)=(-1)^{n-1}\, \frac{\tau(\chi)}{2} \left(\frac{2\pi i}{f}\right)^n \frac{B_{n,\overline{\chi}}}{n!} \,.$

Solving for the Bernoulli number yields

$B_{n,\chi}=(-1)^{n-1}\frac{2 n!}{\tau(\overline{\chi})}\left(\frac{f}{2\pi i}\right)^n L(n,\overline{\chi}) \,.$

This allows us to give decimal approximations for $B_{n,\chi}$ . It remains to compute $B_{n,\chi}$ exactly (i.e., as an algebraic integer). To simplify the above expression, we define

$K_{n,\chi} =(-1)^{n-1}\, 2 n!\left(\frac{f}{2 i}\right)^n$

and write

(3) $B_{n,\chi}=\frac{K_{n,\chi}}{\pi^n\,\tau(\overline{\chi})}\; L(n,\overline{\chi}) \,.$

Note that we can compute $K_{n,\chi}$ exactly in the field $\mathbb{Q}(i)$ .

The following result identifies the denominator of $B_{n,\chi}$ .

Theorem 5.7

Let $n$ and $\chi$ be as above, and define an integer $d$ as follows:

$d = \begin{cases} 1 & \text{if $f$ is divisible by two distinct primes,}\\ 2 & \text{if $f=4$,}\\ 1 & \text{if $f=2^\mu$, $\mu>2$,}\\ np & \text{if $f=p$, $p>2$,}\\ (1-\chi(1+p)) & \text{if $f=p^\mu$, $p>2$, $\mu>1$.}\\ \end{cases}$

Then $dn^{-1}\,B_{n,\chi}$ is integral.

Proof

See [Car59a] for the proof and [Car59b] for further details.

To compute the algebraic integer $d n^{-1} B_{n,\chi}$ , and we compute $L(n,\overline{\chi})$ to very high precision using the Euler product (2) and the formula (3). We carry out the same computation for each of the $\Gal(\Qbar/\Q)$ conjugates of $\chi$ , which by (1) yields the conjugates of $d n^{-1} B_{n,\chi}$ . We can then write down the characteristic polynomial of $d n^{-1} B_{n,\chi}$ to very high precision and recognize the coefficients as rational integers. Finally, we determine which of the roots of the characteristic polynomial is $d n^{-1} B_{n,\chi}$ by approximating them all numerically to high precision and seeing which is closest to our numerical approximation to $d n^{-1} B_{n,\chi}$ . The details are similar to what is explained in Section Fast Computation of Bernoulli Numbers.

Explicit Basis for the Eisenstein Subspace¶

Suppose $\chi$ and $\psi$ are primitive Dirichlet characters with conductors $L$ and $R$ , respectively. Let

(4) $E_{k,\chi,\psi}(q) = c_0 + \sum_{m \geq 1} \left( \sum_{n|m} \psi(n) \cdot \chi(m/n) \cdot n^{k-1}\right) q^{m} \in \Q(\chi, \psi)[[q]],$

where

$c_0 = \begin{cases} 0 & \text{ if } L>1, \\ \ds- \frac{B_{k,\psi}}{2k} & \text{ if } L=1. \end{cases}$

Note that when $\chi=\psi=1$ and $k\geq 4$ , then $E_{k,\chi,\psi} = E_k$ , where $E_k$ is from Chapter Modular Forms.

Miyake proves statements that imply the following in [Miy89, Ch. 7].

Theorem 5.8

Suppose $t$ is a positive integer and $\chi$ , $\psi$ are as above and that $k$ is a positive integer such that $\chi(-1)\psi(-1) = (-1)^k$ . Except when $k=2$ and $\chi=\psi=1$ , the power series $E_{k,\chi,\psi}(q^t)$ defines an element of $M_k(RLt,\chi \psi)$ . If $\chi=\psi=1$ , $k=2$ , $t>1$ , and $E_2(q) = E_{k,\chi,\psi}(q)$ , then $E_2(q) - t E_2(q^t)$ is a modular form in $M_2(\Gamma_0(t))$ .

Theorem 5.9

The Eisenstein series in $M_k(N, \eps)$ coming from Theorem 5.8 with $RLt \mid N$ and $\chi \psi = \eps$ form a basis for the Eisenstein subspace $E_k(N,\eps)$ .

Theorem 5.10

The Eisenstein series $E_{k,\chi,\psi}(q) \in M_k(RL)$ defined above are eigenforms (i.e., eigenvectors for all Hecke operators $T_n$ ). Also $E_2(q) - t E_2(q^t)$ , for $t>1$ , is an eigenform.

Since $E_{k,\chi,\psi}(q)$ is normalized so the coefficient of $q$ is $1$ , the eigenvalue of $T_m$ is the coefficient

$\sum_{n|m} \psi(n) \cdot \chi(m/n) \cdot n^{k-1}$

of $q^m$ (see Proposition 9.10). Also for $f = E_2(q) - t E_2(q^t)$ with $t>1$ prime, the coefficient of $q$ is $1$ , $T_m(f) = \sigma_{1}(m)\cdot f$ for $(m,t)=1$ , and $T_t(f) = ((t+1)-t)f = f$ .

Algorithm 5.11

Given a weight $k$ and a Dirichlet character $\eps$ of modulus $N$ , this algorithm computes a basis for the Eisenstein subspace $E_k(N,\eps)$ of $M_k(N,\eps)$ to precision $O(q^r)$ .

1. [Weight $2$ Trivial Character?] If $k=2$ and $\eps=1$ , output the Eisenstein series $E_2(q) - tE_2(q^t)$ , for each divisor $t\mid N$ with $t\neq 1$ , and then terminate.

[Empty Space?] If $\eps(-1)\neq (-1)^k$ , output the empty list.
[Compute Dirichlet Group] Let $G\set D(N,\Q(\zeta_n))$ be the group of Dirichlet characters with values in $\Q(\zeta_n)$ , where $n$ is the exponent of $(\Z/N\Z)^*$ .
[Compute Conductors] Compute the conductor of every element of $G$ using Algorithm 4.19.
[List Characters $\chi$ ] Form a list $V$ of all Dirichlet characters $\chi \in G$ such that $\cond(\chi)\cdot \cond(\chi/\eps)$ divides $N$ .
[Compute Eisenstein Series] For each character $\chi$ in $V$ , let $\psi = \chi/\eps$ and compute $E_{k,\chi,\psi}(q^t)\pmod{q^r}$ for each divisor $t$ of $N/(\cond(\chi)\cdot \cond(\psi))$ . Here we compute $E_{k,\chi,\psi}(q^t) \pmod{q^r}$ using (4) and Algorithm 5.2.

Remark 5.12

Algorithm 5.11 is what is currently used in Sage. It might be better to first reduce to the prime power case by writing all characters as a product of local characters and combine steps (4) and (5) into a single step that involves orders. However, this might make things more obscure.

Example 5.13

The following is a basis of Eisenstein series for $E_2(\Gamma_1(13))$ .

$f_{1} &= \frac{1}{2} + q + 3q^{2} + 4q^{3} + \cdots,\\ f_{2} &= -\frac{7}{13}\zeta_{12}^{2} - \frac{11}{13} + q + \left(2\zeta_{12}^{2} + 1\right)q^{2} + \left(-3\zeta_{12}^{2} + 1\right)q^{3} + \cdots,\\ f_{3} &= q + \left(\zeta_{12}^{2} + 2\right)q^{2} + \left(-\zeta_{12}^{2} + 3\right)q^{3} + \cdots,\\ f_{4} &= -\zeta_{12}^{2} + q + \left(2\zeta_{12}^{2} - 1\right)q^{2} + \left(3\zeta_{12}^{2} - 2\right)q^{3} + \cdots,\\ f_{5} &= q + \left(\zeta_{12}^{2} + 1\right)q^{2} + \left(\zeta_{12}^{2} + 2\right)q^{3} + \cdots,\\ f_{6} &= -1 + q + -q^{2} + 4q^{3} + \cdots,\\ f_{7} &= q + q^{2} + 4q^{3} + \cdots,\\ f_{8} &= \zeta_{12}^{2} - 1 + q + \left(-2\zeta_{12}^{2} + 1\right)q^{2} + \left(-3\zeta_{12}^{2} + 1\right)q^{3} + \cdots,\\ f_{9} &= q + \left(-\zeta_{12}^{2} + 2\right)q^{2} + \left(-\zeta_{12}^{2} + 3\right)q^{3} + \cdots,\\ f_{10} &= \frac{7}{13}\zeta_{12}^{2} - \frac{18}{13} + q + \left(-2\zeta_{12}^{2} + 3\right)q^{2} + \left(3\zeta_{12}^{2} - 2\right)q^{3} + \cdots,\\ f_{11} &= q + \left(-\zeta_{12}^{2} + 3\right)q^{2} + \left(\zeta_{12}^{2} + 2\right)q^{3} + \cdots.\\$

We computed it as follows:

sage: E = EisensteinForms(Gamma1(13),2)
sage: E.eisenstein_series()

We can also compute the parameters $\chi,\psi,t$ that define each series:

sage: e = E.eisenstein_series()
sage: for e in E.eisenstein_series():
...       print e.parameters()
...
([1], [1], 13)
([1], [zeta6], 1)
([zeta6], [1], 1)
([1], [zeta6 - 1], 1)
([zeta6 - 1], [1], 1)
([1], [-1], 1)
([-1], [1], 1)
([1], [-zeta6], 1)
([-zeta6], [1], 1)
([1], [-zeta6 + 1], 1)
([-zeta6 + 1], [1], 1)

Exercises¶

Exercise 5.1

Suppose $A$ and $B$ are diagonalizable linear transformations of a finite-dimensional vector space $V$ over an algebraically closed field $K$ and that $AB=BA$ . Prove there is a basis for $V$ so that the matrices of $A$ and $B$ with respect to that basis are both simultaneously diagonal.

Exercise 5.2

If $\eps$ is the trivial character of modulus $1$ and $B_k$ are as in Section Examples of Modular Forms of Level 1, then $B_{k,\eps} = B_k$ , except when $k=1$ , in which case $B_{1,\eps} = -B_1 = 1/2$ .

Exercise 5.3

Prove that for $k\geq 2$ if $\eps(-1) \neq (-1)^k$ , then $B_{k,\eps} = 0$ .

Exercise 5.4

Show that the dimension of the Eisenstein subspace $E_3(\Gamma_1(13))$ is $12$ by finding a basis of series $E_{k,\chi,\psi}$ . You do not have to write down the $q$ -expansions of the series, but you do have to figure out which $\chi, \psi$ to use.

Eisenstein Series and Bernoulli Numbers¶

The Eisenstein Subspace¶

Generalized Bernoulli Numbers¶

Algebraically Computing Generalized Bernoulli Numbers¶

Computing Generalized Bernoulli Numbers Analytically¶

Explicit Basis for the Eisenstein Subspace¶

Exercises¶

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Eisenstein Series and Bernoulli Numbers¶

The Eisenstein Subspace¶

Generalized Bernoulli Numbers¶

Algebraically Computing Generalized Bernoulli Numbers¶

Computing Generalized Bernoulli Numbers Analytically¶

Explicit Basis for the Eisenstein Subspace¶

Exercises¶

Table Of Contents

Previous topic

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Purchase a hardcopy of this book from the AMS

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