Previous set of notes: Notes 2. Next set of notes: Notes 4.
On the real line, the quintessential examples of a periodic function are the (normalised) sine and cosine functions
,
, which are
-periodic in the sense that

By taking various polynomial combinations of

and

we obtain more general trigonometric polynomials that are

-periodic; and the theory of Fourier series tells us that all other

-periodic functions (with reasonable integrability conditions) can be approximated in various senses by such polynomial combinations. Using Euler’s identity, one can use

and

in place of

and

as the basic generating functions here, provided of course one is willing to use complex coefficients instead of real ones. Of course, by rescaling one can also make similar statements for other periods than

.

-periodic functions

can also be identified (by abuse of notation) with functions

on the quotient space

(known as the
additive
-torus or
additive unit circle), or with functions
![{f: [0,1] \rightarrow {\bf C}}](https://s0.wp.com/latex.php?latex=%7Bf%3A+%5B0%2C1%5D+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
on the fundamental domain (up to boundary)
![{[0,1]}](https://s0.wp.com/latex.php?latex=%7B%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
of that quotient space with the periodic boundary condition

. The map

also identifies the additive unit circle

with the
geometric unit circle 
, thanks in large part to the fundamental trigonometric identity

; this can also be identified with the
multiplicative unit circle 
. (Usually by abuse of notation we refer to all of these three sets simultaneously as the “unit circle”.) Trigonometric polynomials on the additive unit circle then correspond to ordinary polynomials of the real coefficients

of the geometric unit circle, or Laurent polynomials of the complex variable

.
What about periodic functions on the complex plane? We can start with singly periodic functions
which obey a periodicity relationship
for all
in the domain and some period
; such functions can also be viewed as functions on the “additive cylinder”
(or equivalently
). We can rescale
as before. For holomorphic functions, we have the following characterisations:
Proposition 1 (Description of singly periodic holomorphic functions)
In both cases, the coefficients
can be recovered from
by the Fourier inversion formula 
for any
in
(in case (i)) or
(in case (ii)).
Proof: If
is
-periodic, then it can be expressed as
for some function
on the “multiplicative cylinder”
, since the fibres of the map
are cosets of the integers
, on which
is constant by hypothesis. As the map
is a covering map from
to
, we see that
will be holomorphic if and only if
is. Thus
must have a Laurent series expansion
with coefficients
obeying (2), which gives (1), and the inversion formula (5) follows from the usual contour integration formula for Laurent series coefficients. The converse direction to (i) also follows by reversing the above arguments.
For part (ii), we observe that the map
is also a covering map from
to the punctured disk
, so we can argue as before except that now
is a bounded holomorphic function on the punctured disk. By the Riemann singularity removal theorem (Exercise 35 of 246A Notes 3)
extends to be holomorphic on all of
, and thus has a Taylor expansion
for some coefficients
obeying (4). The argument now proceeds as with part (i).
The additive cylinder
and the multiplicative cylinder
can both be identified (on the level of smooth manifolds, at least) with the geometric cylinder
, but we will not use this identification here.
Now let us turn attention to doubly periodic functions of a complex variable
, that is to say functions
that obey two periodicity relations

for all

and some periods

, which to avoid degeneracies we will assume to be linearly independent over the reals (thus

are non-zero and the ratio

is not real). One can rescale

by a common scaling factor

to normalise either

or

, but one of course cannot simultaneously normalise both parameters in this fashion. As in the singly periodic case, such functions can also be identified with functions on the additive

-torus

, where

is the lattice

, or with functions

on the solid parallelogram bounded by the contour

(a fundamental domain up to boundary for that torus), obeying the boundary periodicity conditions

for

in the edge

, and

for

in the edge

.
Within the world of holomorphic functions, the collection of doubly periodic functions is boring:
Proposition 2 Let
be an entire doubly periodic function (with periods
linearly independent over
). Then
is constant.
In the language of Riemann surfaces, this proposition asserts that the torus
is a non-hyperbolic Riemann surface; it cannot be holomorphically mapped non-trivially into a bounded subset of the complex plane.
Proof: The fundamental domain (up to boundary) enclosed by
is compact, hence
is bounded on this domain, hence bounded on all of
by double periodicity. The claim now follows from Liouville’s theorem. (One could alternatively have argued here using the compactness of the torus
.
To obtain more interesting examples of doubly periodic functions, one must therefore turn to the world of meromorphic functions – or equivalently, holomorphic functions into the Riemann sphere
. As it turns out, a particularly fundamental example of such a function is the Weierstrass elliptic function 
which plays a role in doubly periodic functions analogous to the role of
for
-periodic real functions. This function will have a double pole at the origin
, and more generally at all other points on the lattice
, but no other poles. The derivative 
of the Weierstrass function is another doubly periodic meromorphic function, now with a triple pole at every point of
, and plays a role analogous to
. Remarkably, all the other doubly periodic meromorphic functions with these periods will turn out to be rational combinations of
and
; furthermore, in analogy with the identity
, one has an identity of the form 
for all
(avoiding poles) and some complex numbers
that depend on the lattice
. Indeed, much as the map
creates a diffeomorphism between the additive unit circle
to the geometric unit circle
, the map
turns out to be a complex diffeomorphism between the torus
and the elliptic curve

with the convention that

maps the origin

of the torus to the point

at infinity. (Indeed, one can view elliptic curves as “multiplicative tori”, and both the additive and multiplicative tori can be identified as smooth manifolds with the more familiar
geometric torus, but we will not use such an identification here.) This fundamental identification with elliptic curves and tori motivates many of the further remarkable properties of elliptic curves; for instance, the fact that tori are obviously an abelian group gives rise to an abelian group law on elliptic curves (and this law can be interpreted as an analogue of the trigonometric sum identities for

). The description of the various meromorphic functions on the torus also helps motivate the more general
Riemann-Roch theorem that is a fundamental law governing meromorphic functions on other compact Riemann surfaces (and is discussed further in
these 246C notes). So far we have focused on studying a single torus

. However, another important mathematical object of study is the space of
all such tori, modulo isomorphism; this is a basic example of a
moduli space, known as the
(classical, level one) modular curve 
. This curve can be described in a number of ways. On the one hand, it can be viewed as the upper half-plane

quotiented out by the discrete group

; on the other hand, by using the

-invariant, it can be identified with the complex plane

; alternatively, one can compactify the modular curve and identify this compactification with the Riemann sphere

. (This identification, by the way, produces a very short proof of the little and great Picard theorems, which we proved in
246A Notes 4.) Functions on the modular curve (such as the

-invariant) can be viewed as

-invariant functions on

, and include the important class of
modular functions; they naturally generalise to the larger class of (weakly)
modular forms, which are functions on

which transform in a very specific way under

-action, and which are ubiquitous throughout mathematics, and particularly in number theory. Basic examples of modular forms include the
Eisenstein series, which are also the Laurent coefficients of the Weierstrass elliptic functions

. More number theoretic examples of modular forms include (suitable powers of)
theta functions 
, and the
modular discriminant 
. Modular forms are

-periodic functions on the half-plane, and hence by Proposition
1 come with Fourier coefficients

; these coefficients often turn out to encode a surprising amount of number-theoretic information; a dramatic example of this is the famous
modularity theorem, (a special case of which was) used amongst other things to establish Fermat’s last theorem. Modular forms can be generalised to other discrete groups than

(such as
congruence groups) and to other domains than the half-plane

, leading to the important larger class of
automorphic forms, which are of major importance in number theory and representation theory, but which are well outside the scope of this course to discuss.
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