Math 252: Modular Abelian Varieties

Let D be the fundamental domain for G=PSL_{2}(Z)
that I described in the last lecture, which is illustrated below:

Last time we proved that if *z *is any point in the upper
half plane, then there is an element *g* in G such that *g*(*z*)
is in D. The proof was nonconstructive. Here is an algorithm to find such a
*g* in practice.

Do the following until |

This process must terminate, because in step 2 the imaginary part
of -1/*z* is bigger than
the imaginary part of *z*, and the proof from last time showed that the
imaginary parts of the conjugates of *z* are bounded above.

**Remarks.**

- John Cremona makes the following remark when describing this algorithm in
Section 2.14 of his book. "In practice one must be careful about rounding
errors, as it is quite possible to have both |
*z*|<1 and |-1/*z*|<1 after rounding, which is liable to prevent the algorithm from terminating." So be careful if you implement this algorithm. - This is almost the same as the algorithm for computing the continued fraction expansion of a real number! Explore this further on your own if you want.

Note that this algorithm also gives an algorithm to write an arbitrary element of G in terms of

and .

For example, suppose . Hit 2i by g, then find a combination of S and T that conjugates g back into D. We get

**Remark. ** If you wanted to do computations like this efficiently,
you would obtain the representation of g from the partial convergents of the
continued fraction of a/b, where a and b are the entries in the first column
of g. See the end of Section 2.1 of Cremona's book for enough of a hint as to
what to do.

As you will show in your homework reduction modulo N is surjective, so we have an exact sequence

**DEFINITION: **The group
is by definition the kernel of the reduction map. It is called the **principal
congruence subgroup** of level N. A **congruence subgroup**
is any subgroup of SL_{2}(**Z**) that contains
for some N.

It is clear that any congruence subgroup has finite index, since
does. What about
the converse? This is the so-called "congruence subgroup problem."
According to this MathSciNet review, if p
is a prime, then every finite index subgroup of SL_{2}(**Z**[1/p])
is a congruence subgroup, and for any n>2, *all* finite index subgroups
of SL_{2}(**Z**) are congruence subgroups. However, there
are plenty of finite index subgroups of SL_{2}(**Z**) that
are not congruence subgroups. This
paper by Hsu contains an algorithm to decide if a finite index subgroup
is a congruence subgroup, and gives an example of a subgroup of index 12 that
it is not a congruence subgroup. Also this MathSciNet
review of a paper by Thompson is about his proof of a surprising conjecture
of Atkin about modular forms for noncongruence subgroups.

Since is a kernel, it is a normal subgroup. There are two other extremely popular families of congruence subgroups, neither of which are normal:

Helena Verrill's fundamental domains program draws fundamental domains. You should probably look through her description of the interesting mathematical algorithms used in the program. Here are three screen shots, which illustrates some fundamental domains.

The program works by finding right coset representatives for the congruence
subgroup in SL_{2}(**Z**), and translating around a fundamental
domain for SL_{2}(**Z**). The coset representatives are
chosen, as described in Verrrill's paper, so
that the domain is connected and not tiny -- it wouldn't be interesting to draw
a fundamental that consists of a few dots on the screen, even though it might
be accurate!

Finding coset representatives for any of the three families of congruence subgroups
mentioned above is relatively straightforward. To make the problem easy to think
about, note that you can quotient out by
first. Then the question amounts to finding coset representatives for a subgroup
of SL_{2}(**Z**), which is reasonably straightforward.
For example:

**Proposition. ***The coset representatives for Gamma_0(N)
are in natural bijection with the elements of ***P**^{1}(**Z***/N***Z**)*.*

Let G be a finite index subgroup of SL_{2}(**Z**). Let
R be a set of coset representatives for G. Define maps s and t from R to G as
follows. If r in R then there exists a unique r' in R such that GrS = Gr'. Let
s(r) = rS(r')^(-1). Likewise, there is a unique r' such that GrT = Gr' and we
let t(r) = rT(r')^(-1). Note that s(r) and t(r) are in G for all r.

**Proposition. ***G is generated by the union of s(R) and t(R).*

Proof. Omited.

There are much more sophisticated algorithms that use group actions on trees. Verrill implemented much of this in MAGMA, but unfortunately I don't have time to talk about it today.

Next week we will consider the quotient of the upper half plane by the action
of a congruence subgroup. This quotient space is a noncompact Riemann surface.
The missing cusps are in bijection with the orbits of the action of the congruence
subgroup on **P**^{1}(**Q**). Adding them
in we obtain a **compact Riemann surface**. On Monday we'll construct
this Riemann surface, and discuss some of its properties. On Wednesday, we'll
talk about how to represent its homology using **modular symbols.**
On Friday, we'll describe a **finite presentation for homology that is
due to Yuri Manin**, and sketch his proof that the presentation is correct.