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) =
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
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
(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
E6(z) = 1 − 504
generate the algebra of all the modular forms on SL2(Z).