6 1. BASIC FACTS

Hence, E2(z) is not a modular form. Although E2(z) is not a modular form, it plays

an important role in the subject. For example, E2(z) is required when studying

differential operators on modular forms (see Section 2.3). It also plays an important

role in the theory of p-adic modular forms [Se3].

The coeﬃcients of Eisenstein series satisfy many congruences which will prove

to be important throughout this monograph. First we briefly define the notion of

a congruence between q-series.

Definition 1.21. Suppose that

F (q) =

n≥n0

a(n)qn,

G(q) =

n≥m0

b(n)qn

are q-series with coeﬃcients in a commutative ring A, and suppose that m ⊂ A is

an ideal. We say that F is congruent to G modulo m if a(n) − b(n) ∈ m for every

n. We denote this by

F (q) ≡ G(q) (mod m),

Here we record elementary congruences which follow from classical facts on

Bernoulli numbers. We also identify certain trivial zeros for some of these Eisenstein

series.

Lemma 1.22. Suppose that k ≥ 2 is even.

(1) We have Ek(z) ≡ 1 (mod 24).

(2) If p is prime and (p − 1) | k, then Ek(z) ≡ 1 (mod

pordp(2k)+1).

(3) If 4 ≤ k ≡ 0 (mod 3), then Ek(ω) = 0.

(4) If 4 ≤ k ≡ 2 (mod 4), then Ek(i) = 0.

(5) If p ≥ 3 is prime, then

Ep+1(z) ≡ E2(z) (mod p).

Sketch of the proof. Since z = i (resp. z = ω) is fixed by the modular

transformation Sz = −1/z (resp. Az = −(z + 1)/z), the definition of a modular

form implies that Ek(i) = 0 whenever k ≡ 2 (mod 4), and Ek(ω) = 0 whenever

k ≡ 0 (mod 3). The claimed congruences follow immediately from the definition of

the Eisenstein series and the von Staudt-Clausen theorem on the divisibility of de-

nominators of Bernoulli numbers, and the Kummer congruences between Bernoulli

numbers (see Chapter 15 of [IR]).

A fundamental fact is that the two Eisenstein series

E4(z) = 1 + 240

∞

n=1

σ3(n)qn,

E6(z) = 1 − 504

∞

n=1

σ5(n)qn

generate the algebra of all the modular forms on SL2(Z).