forms for T0(k). Part of the extra difficulties involves the presence of tor-
sion in these groups. We need more detailed analysis, in this and subsequent
chapters, to handle these richer groups.
Chapter 4 is a systematic study of theta identities. Theta constant
identities are interesting for several reasons. One reason is their inherent
elegance and symmetry. There is something tantalizingly beautiful about
the identities of Jacobi, for example,
" 0 '
" 0 "
" 0 '
" 0 "
' 1 "
0 _
" 1 "
or its restriction to z = 0 (known as the Jacobi quartic identity)
" 0 "
(o,r) =
' o"
' 1 "
Aside from the inherent beauty of the form there is a combinatorial content
to the identity. It relates the number of representations of an integer as a
sum of four squares to its representation as a sum of four triangular numbers.
This is of course just the beginning of a chapter. As one delves deeper into
the theory, one finds more and more beautiful identities with more and more
combinatorial content.
We present four distinct ways of constructing such identities. In two
of theses methods, we use the classical technique of constructing finite di-
mensional linear spaces of theta functions or modular forms or functions on
certain Riemann surfaces and use linear algebra and the simple notions of
independence and dependence. We present another newer technique which
makes use of the fact that we can use theta functions to construct elliptic
functions and the fact that the sum of the residues of an elliptic function
in a period parallelogram vanishes. This technique is very powerful and
succeeds in giving a rather large set of identities. The main idea here is to
construct the correct elliptic function, which turns out to be more an art
than a science, that leads to an interesting identity. The fourth method for
constructing identities uses uniformizations of Riemann surfaces. In a very
nontrivial sense the next two chapters are also studies of theta identities,
this time of a very special sort with a very special purpose.
In Chapter 5 we turn to the congruences discovered by Ramanujan for
the partition function and show how they follow in a rather simple way from
function theory on the appropriate Riemann surfaces. The main ingredient
is the construction of the same function in more than one way. Some of
the constructions involve averaging operators. It turns out that the aver-
aging processes produce in some cases constant functions. We study the
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