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Fearless Symmetry by Robert Gross, Avner Ash

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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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