240 CHAPTER 21

a

3

= 1 + 3 − 5 =−1. One more: E(F

5

) ={O, (0, 0), (0, 4), (1, 0), (1, 4)},

and so a

5

= 1 + 5 − 5 = 1.

9

We can also take the following inﬁnite product and expand it to

as many terms as we like:

q

∞

!

n=1

(1 − q

n

)

2

(1 − q

11n

)

2

= q(1 − q)

2

(1 − q

2

)

2

(1 − q

3

)

2

(1 − q

4

)

2

···

(1 − q

11

)

2

(1 − q

22

)

2

(1 − q

33

)

2

(1 − q

44

)

2

···

= q + b

2

q

2

+ b

3

q

3

+ b

4

q

4

+ b

5

q

5

+···

= q − 2q

2

− q

3

+ 2q

4

+ q

5

+···. (21.2)

This power series in q is a cuspidal normalized newform of level 11.

The amazing thing is that b

= a

for any prime other than 11! In

other words, to compute the number of solutions to y

2

+ y ≡ x

3

− x

2

(mod ), you can multiply the above product up to and including

the terms containing q

, ﬁgure out the coefﬁcient of q

, and use that

number to compute #E(F

). You can see that this is true for = 2, 3,

and 5 from our preceding computations.

A vast generalization of this is the conjecture that there are

(similar but more complicated) reciprocity laws between any Galois

representation coming from the

´

etale cohomology of a Z-variety

and an appropriate generalization of a modular form, called an

“automorphic representation.” (An automorphic representation is

indeed a kind of group representation, but with a totally different

source and target from a Galois representation.) This conjecture

is part of a grand system of conjectures known as the “Langlands

program,” after the Canadian mathematician Robert Langlands.

A Physical Analogy

To conclude, we can draw an analogy between the linear repre-

sentations of the absolute Galois group G of Q and old-fashioned

relativistic particle physics.

9

It is just a coincidence that #E(F

) = 5 for = 2, 3, and 5.

A LAST LOOK AT RECIPROCITY 241

In physics, the key group is L, the Lorentz group of spacetime

coordinate transformations. All physical laws must be invariant

under L. In quantum mechanics, a system is given by a large set

of complex vectors H (actually inﬁnite-dimensional), and a state

S of the system is described by x

S

, a nonzero vector in H (or

more accurately, a line in H). If you act on your system by a

transformation of coordinates in spacetime, this changes the way

you describe the given state of the system, but not the actual

physical state itself. So this change of coordinates must take the

vector x

S

to a new vector that describes the same state S in the new

coordinates. The laws of quantum mechanics say that this gives a

linear action of L on H. That representation is the system from

this point of view. If the representation cannot be decomposed

linearly into simpler ones, we call it “irreducible.” The irreducible

representations correspond to elementary particles. In this analogy,

a reciprocity law in physics would be given by a black box that

enables you to compute the observables of the system (such as the

energy levels) given the representation of L.

Similarly, the absolute Galois group G has many linear repre-

sentations, and each one reveals or describes a number-theoretical

system. The irreducible representations then play the role of

particles.

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