236 CHAPTER 21

As our last example of a black box, we consider modular forms.

Although modular forms were discovered or invented long before

´

etale cohomology and have a lot of different applications in math-

ematics and physics, they can also be used in reciprocity laws

in conjunction with the

´

etale cohomology of certain Z-varieties.

Although the theory of

´

etale cohomology gives us a huge realm

to think about, often we cannot say much unless we get another

type of gadget, much more explicit and easy to work with, that

in particular situations contains the same information. Besides

modular forms, other such black boxes are ﬁnite ﬂat group schemes

and formal groups.

4

Modular Forms

Modular forms are explained in various books about Wiles’s proof of

Fermat’s Last Theorem (Hellegouarch, 2002; Singh, 1997; van der

Poorten, 1996). For our purposes, you can think of a modular form

as an inﬁnite series in powers of a variable q: a

0

+ a

1

q + a

2

q

2

+···.

Not every such inﬁnite series is a modular form. The coefﬁcients a

i

have to be formed according to certain rules. The a

i

are called the

Fourier coefﬁcients of the modular form because the series in q can

be thought of as the Fourier series of a certain function. (The use of

the letter q here is traditional and it should not be confused with a

prime number. Therefore, in this chapter we will use to denote a

varying prime number.)

There are particular kinds of modular forms called cuspidal

normalized newforms. These all have a

0

= 0, a

1

= 1, and lots of

powerful properties.

5

Here is the great fact:

FACT: Every cuspidal normalized newform gives a

reciprocity law.

We now explain what this reciprocity law looks like, and then we

end with some examples.

4

Notice the word “group” keeps cropping up. Much of what can be done can only be done

when there is a group law around.

5

For example, if m and n share no common prime factor, then a

mn

= a

m

a

n

.

A LAST LOOK AT RECIPROCITY 237

Because modular forms are related to

´

etale cohomology, you can

guess that there will be a whole series of Galois representations in

the picture, one for each prime p. Start with a particular prime p.

THEOREM 21.1: If q + a

2

q

2

+ a

3

q

3

+··· is a cuspidal

normalized newform, then there exist

1. a positive integer N, called the level of the newform;

2. a ﬁeld k that contains Q

p

and also all the coefﬁcients a

i

;

3. a two-dimensional linear Galois representation

r : G → GL(2, k);

which obey the following rule: If is any prime that does

not evenly divide pN, then r is unramiﬁed at , and

χ

r

(Frob

) = a

.

6

If you are worried about Q

p

and k, we can state a simpler

corollary of this theorem, more along the lines of our earlier

exposition:

COROLLARY: Given the cuspidal normalized newform as

above, assume that all of the a

i

’s are ordinary integers. Then

there exists a two-dimensional linear mod p Galois

representation r : G → GL(2, F

p

) with the property that

χ

r

(Frob

) ≡ a

(mod p).

What should we think about this? The point is that it is not

difﬁcult to write down zillions of examples of cuspidal normalized

newforms. They, and Theorem 21.1, then immediately tell you

that there are zillions of different two-dimensional linear rep-

resentations of the absolute Galois group G of Q. This means,

among other things, that G is very big. And yet you may hope

that at least all of its two-dimensional linear representations

come about like this. That is not true, yet there is an important

6

Although we know from this theorem that the Galois representations in (3) exist, we

cannot write them down explicitly, because we do not have a good understanding of the

elements of the absolute Galois group of Q, one by one.

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