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

on the fundamental domain (up to boundary)

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.

** — 1. Doubly periodic functions — **

Throughout this section we fix two complex numbers that are linearly independent over , which then generate a lattice .

We now study the doubly periodic meromorphic functions with respect to these periods that are not identically zero. We first observe some constraints on the poles of these functions. Of course, by periodicity, the poles will themselves be periodic, and thus the set of poles forms a finite union of disjoint cosets of the lattice . Similarly, the zeroes form a finite union of disjoint cosets . Using the residue theorem, we can obtain some further constraints:

**Lemma 3 (Consequences of residue theorem)** Let be a doubly periodic meromorphic function (not identically zero) with periods , poles at , and zeroes at .
- (i) The sum of residues at each (i.e., we sum one residue per coset) is equal to zero.
- (ii) The number of poles (counting multiplicity, but only counting once per coset) is equal to the number of zeroes (again counting multiplicity, and once per coset).
- (iii) The sum of the poles (counting multiplicity, and working in the group ) is equal to the sum of the zeroes .

*Proof:* For (i), we first apply a translation so that none of the pole cosets intersects the fundamental parallelogram boundary ; this of course does not affect the sum of residues. Then, by the residue theorem, the sum in (i) is equal to the expression

(the sign depends on whether this contour is oriented counter-clockwise or clockwise). But from the double periodicity we see that the integral vanishes (the contribution of parallel pairs of edges cancel each other). For part (ii), apply part (i) to the logarithmic derivative , which is also doubly periodic.
For part (iii), we again translate so that none of the pole or zero cosets intersects , noting from part (ii) that any such translation affects the sum of poles and sum of zeroes by the same amount. By the residue theorem, it now suffices to show that

lies in the lattice . But one can rewrite this using the double periodicity as
so it suffices to show that is an integer for . But (a slight modification of) the argument principle shows that this number is precisely the winding number around the origin of the image of under the map , and the claim follows.
This lemma severely limits the possible number of behaviors for the zeroes and poles of a meromorphic function. To formalise this, we introduce some general notation:

**Definition 4 (Divisors)**
- (i) A divisor on the torus is a formal integer linear combination , where ranges over a finite collection of points in the torus (i.e., a finite collection of cosets ), and are integers, with the obvious additive group structure; equivalently, the space of divisors is the free abelian group with generators for (with the convention ).
- (ii) The number is the
*degree* of a divisor , the point is the *sum* of , and each is the *order* of the divisor at (with the convention that the order is if does not appear in the sum). A divisor is *non-negative* (or *effective*) if for all . We write if is non-negative (i.e., the order of is greater than or equal to that of at every point , and if and . - (iii) Given a meromorphic function (or equivalently, a doubly periodic function ) that is not identically zero, the
*principal divisor* is the divisor , where ranges over the zeroes and poles of , and is the order of the zero (if is a zero) or negative the order of the pole (if is a pole). - (iv) Given a divisor , we define to be the space of all meromorphic functions that are either zero, or are such that . That is to say, consists of those meromorphic functions that have at most a pole of order at if is positive, or at least zero of order if is negative.

A divisor can be viewed as an abstraction of the concept of a set of zeroes and poles (counting multiplicity). Observe that principal divisors obey the laws , when are meromorphic and non-zero. In particular, the space of principal divisors is a subgroup of the space of all divisors. By Lemma 3(ii), all principal divisors have degree zero, and from Lemma 3(iii), all principal divisors have sum zero as well. Later on we shall establish the converse claim that every divisor of degree and sum zero is a principal divisor; see Exercise 7.

**Remark 5** One can define divisors on other Riemann surfaces, such as the complex plane . Observe from the fundamental theorem of algebra that if one has two non-zero polynomials , then if and only if divides as a polynomial. This may give some hint as to the origin of the terminology “divisor”. The machinery of divisors turns out to have a rich algebraic and topological structure when applied to more general Riemann surfaces than tori, for instance enabling one to associate an abelian variety (the Jacobian variety) to every algebraic curve; see these 246C notes for further discussion.

It is easy to see that is always a vector space. All non-zero meromorphic functions belong to at least one of the , namely , so to classify all the meromorphic functions on , it would suffice to understand what all the spaces are.

Liouville’s theorem (in the form of Proposition 2) tells us that all elements of – that is to say, the holomorphic functions on – are constant; thus is one-dimensional. If is a negative divisor, the elements of are thus constant and have at least one zero, thus in these cases is trivial.

Now we gradually work our way up to higher degree divisors . A basic fact, proven from elementary linear algebra, is that every time one adds a pole to , the dimension of the space only goes up by at most one:

**Lemma 6** For any divisor and any , is a subspace of of codimension at most one. In particular, is finite-dimensional for any .

*Proof:* It is clear that is a subspace of . If has order at , then there is a linear functional that assigns to each meromorphic function the coefficient of the Laurent expansion of at (note from periodicity that the exact choice of coset representative is not relevant. A little thought reveals that the kernel of is precisely , and the first claim follows. The second claim follows from iterating the first claim, noting that any divisor can be obtained from a suitable negative divisor by the addition of finitely many poles .

Now consider the space for some point . Lemma 6 tells us that the dimension of this space is either one or two, since was one-dimensional. The space consists of functions that possibly have a simple pole at most at , and no other poles. But Lemma 3(i) tells us that the residue at has to vanish, and so is in fact in and thus is constant. (One could also argue here using the other two parts of Lemma 2; how?) So is no larger than , and is thus also one-dimensional.

Now let us study the space – the space of meromorphic functions that have at most a double pole at and no other poles. Again, Lemma 6 tells us that this space is one or two dimensional. To figure out which, we can normalise to be the origin coset . The question is now whether there is a doubly periodic meromorphic function that has a double pole at each point of . A naive candidate for such a function would be the infinite series

however this series turns out to not be absolutely convergent. Somewhat in analogy with the discussion of the Weierstrass and Hadamard factorisation theorems in

Notes 1, we then proceed instead by working with the normalised function

defined by the formula

(6). Let us first verify that the series in

(6) is absolutely convergent for

. There are only finitely many

with

, and all the summands are finite for

, so we only need to establish convergence of the tail

However, from the fundamental theorem of calculus we have

so to demonstrate absolute convergence it suffices to show that

But a simple volume packing argument (considering the areas of the translates

of the fundamental domain

) shows that the number of lattice points

in any disk

,

is

, and so by dyadic decomposition as in

Notes 1, the series is absolutely convergent. Further repetition of the arguments from

Notes 1 shows that the series in

(6) converges locally uniformly in

, and thus is holomorphic on this set. Furthermore, for any

, the same arguments show that

stays bounded in a punctured neighbourhood of

, thus by the Riemann singularity removal theorem

is equal to

plus a bounded holomorphic function in the neighbourhood of

. Thus

is meromorphic with double poles (and vanishing residue) at every lattice point

, and no other poles.

Now we show that is doubly periodic, thus and for . We just prove the first identity, as the second is analogous. From (6) we have

The series on the right is absolutely convergent, and on every coset of

it telescopes to zero. The claim then follows by Fubini’s theorem.

By construction, lies in , and is clearly non-constant. Thus is two-dimensional, being spanned by the constant function and . By translation, we see that is two-dimensional for any other point as well.

From (6) it is also clear that the function is even: . In particular, for any avoiding the half-lattice (so that and occupy different locations in the torus ), the function has a zero at both and . By Lemma 3(ii) there are no other zeroes of this function (and this claim is also consistent with Lemma 3(iii)); thus the divisor of this function is given by

If

lies in the half-lattice

but not in

(thus, it lies in one of the

*half-periods* ,

, or

) then from the even and doubly periodic nature of

we see that

for all

, so

in fact must have at least a double zero at

, and again from Lemma

3(ii) these are the only zeroes of this function. So the identity

(9) also holds in this case.

**Exercise 7 (Classification of principal divisors)**

Now let us study the space , where we again normalise for sake of discussion. Lemma 6 tells us that this space is two or three dimensional, being spanned by , , and possibly one other function. Note that the derivative of the meromorphic function is also doubly periodic with a triple pole at , so it lies in and is not a linear combination of or (as these have a lower order singularity at ). Thus is three-dimensional, being spanned by . A formal term-by-term differentiation of (6) gives (7). To justify (7), observe that the arguments that demonstrated the meromorphicity of the right-hand side of (6) also show the meromorphicity of (7). From Fubini’s theorem, the fundamental theorem of calculus, and (6) we see that

for any contour

in

from one point

to another

, and the claim

(7) now follows from another appeal to the fundamental theorem of calculus. Of course,

will then also be three-dimensional for any other point

on the torus. From

(7) we also see that

is odd; this also follows from the even nature of

. From the oddness and periodicity

has to have zeroes at the half-periods

; in particular, from Lemma

3(ii) there are no other zeroes, and the principal divisor is given by

Turning now to , we could differentiate yet again to generate a doubly periodic function with a fourth order pole at the origin, but we can also work with the square of the Weierstrass function. From Lemma 6 we conclude that is four-dimensional and is spanned by . In a similar fashion, is a five-dimensional space spanned by .

Something interesting happens though at . Lemma 6 tells us that this space is the span of , and possibly one other function, which will have a pole of order six at the origin. Here we have *two* natural candidates for such a function: the cube of the Weierstrass function, and the square of its derivative. Both have a pole of order exactly six and lie in , and so must be a linear combination of . But since are even and are odd, must in fact just be a linear combination of . To work out the precise combination, we see by repeating the derivation of (7) that

for any

, so that the function

, which extends holomorphically to the origin, has

derivative at the origin equal to

, where the

Eisenstein series is defined by the formula

we can also extend this to the

case with convention

. Note from the symmetric nature of

that

vanishes for odd

; this is consistent with the even nature of

. We thus have the Laurent expansion

for

near zero. This then gives the further Laurent expansions

From these expansions we see that the decomposition of

into a linear combination of

does not actually involve

(as this function is the only one that has a

term in its Laurent expansion), and on comparing

coefficients we see that the coefficient of

must be

. Thus we have a linear relationship of the form

(8) for some coefficients

, which on inspection of the

and constant terms leads to the formulae

**Exercise 8** Derive (8) directly from Proposition 2 by showing that the difference between the two sides is doubly periodic and holomorphic after removing singularities.

**Exercise 9 (Classification of doubly periodic meromorphic functions)**
- (i) For any , show that has dimension , and every element of this space is a polynomial combination of .
- (ii) Show that every doubly periodic meromorphic function is a rational function of .

We have an alternate form of (8):

**Exercise 10** Define the roots , , .

If we now define the elliptic curve

to be the union of a certain cubic curve in the complex plane

together with the point at infinity (where the notions of “curve” and “plane” are relative to the underlying complex field

rather than the more familiar real field

), then we have a map

from

to

, with the convention that the origin

is mapped to the point at infinity

. For instance, the half-periods

,

,

are mapped to the points

of

respectively.

**Lemma 11** The map defined by (11) is a bijection between and .

Among other things, this lemma implies that the elliptic curve is topologically equivalent (i.e., homeomorphic to) a torus, which is not an entirely obvious fact (though if one squints hard enough, the real analogue of an elliptic curve does resemble a distorted slice of a torus embedded in ).

*Proof:* Clearly is the only point that maps to , and (from (10)) the half-periods are the only points that map to . It remains to show that all the other points arise via from exactly one element of . The function has exactly two zeroes by Lemma 3(ii), which lie at for some as is even; since , is not equal to , hence is not a half-period. As is odd, the map (11) must therefore map to the two points of the elliptic curve that lie above , and the claim follows.

Analogously to the Riemann sphere , the elliptic curve can be given the structure of a Riemann surface, by prescribing the following charts:

- (i) When is a point in other than or , then locally is the graph of a holomorphic branch of the square root of near , and one can use as a coordinate function in a sufficiently small neighbourhood of .
- (ii) In the neighbourhood of for some , the function has a simple zero at and so has a local inverse that maps a neighbourhood of to a neighbourhood of , and a point sufficiently near can be parameterised by . One can then use as a coordinate function in a neighbourhood of .
- (iii) A neighbourhood of consists of and the points in the remaining portion of with sufficiently large; then is asymptotic to a square root of , so in particular and should both go to zero as goes to infinity in . We rewrite the defining equation of the curve in terms of and as . The function has a simple zero at zero and thus has a holomorphic local inverse that maps to , and we have in a neighbourhood of infinity. We can then use as a coordinate function in a neighbourhood of , with the convention that this coordinate function vanishes at infinity.

It is then a tedious but routine matter to check that has the structure of a Riemann surface. We then claim that the bijection defined by (11) is holomorphic, and thus a complex diffeomorphism of Riemann surfaces. In the neighbourhood of any point of the torus other than the origin , maps to a neighbourhood of finite point of , including the three points , the holomorphicity is a routine consequence of composing together the various local holomorphic functions and their inverses. In the neighbourhood of the origin , maps for small to a point of with a Laurent expansion

from the Laurent expansions of

, so in particular the coordinate

takes the form

where the error term

is holomorphic, with

mapping to

. In particular the map is a local complex diffeomorphism here and again we have holomorphicity. We thus conclude that the elliptic curve

is complex diffeomorphic to the torus

using the map

. From Exercise

9, the meromorphic functions on

may be identified with the rational functions on

.

While we have shown that all tori are complex diffeomorphic to elliptic curves, the converse statement that all elliptic curves are diffeomorphic to tori will have to wait until the next section for a proof, once we have set up the machinery of modular forms.

**Exercise 12 (Group law on elliptic curves)**
- (i) Let be three distinct elements of the torus that are not equal to the origin . Show that if and only if the three points , , are collinear in , in the sense that they lie on a common complex line for some complex numbers with not both zero.
- (ii) What happens in (i) if (say) and agree? What about if ?
- (iii) Using (i), (ii), give a purely geometric definition of a group addition law on the elliptic curve which is compatible with the group addition law on the torus via (11). (We remark that the associativity property of this law is not obvious from a purely geometric perspective, and is related to the Cayley-Bacharach theorem in classical geometry; see this previous blog post.)

**Exercise 13 (Addition law)** Show that for any lying in distinct cosets of , one has

**Exercise 14 (Special case of Riemann-Roch)**

**Exercise 15 (Elliptic integrals)**

**Remark 16** The integral is an example of an elliptic integral; many other elliptic integrals (such as the integral arising when computing the perimeter of an ellipse) can be transformed into this form (or into a closely related integral) by various elementary substitutions. Thus the Weierstrass elliptic function can be employed to evaluate elliptic integrals, which may help explain the terminology “elliptic” that occurs throughout these notes. In 246C notes we will introduce the notion of a meromorphic -form on a Riemann surface. The identity (12) can then be interpreted in this language as the differential form identity , where are the standard coordinates on the elliptic curve ; the meromorphic -form is initially only defined on outside of the four points , but this identity in fact reveals that the form extends holomorphically to all of ; it is an example of what is known as an Abelian differential of the first kind.

**Remark 17** The elliptic curve (for various choices of parameters ) can be defined in other fields than the complex numbers (though some technicalities arise in characteristic two and three due to the pathological behaviour of the discriminant in those cases). On the other hand, the Weierstrass elliptic function is a transcendental function which only exists in complex analysis and does not have a direct analogue in other fields. So this connection between elliptic curves and tori is specific to the complex field. Nevertheless, many facts about elliptic curves that were initially discovered over the complex numbers through this complex-analytic link to tori, were then reproven by purely algebraic means, so that they could be extended without much difficulty to many other fields than the complex numbers, such as finite fields. (For instance, the role of the complex torus can be replaced by the Jacobian variety, which was briefly introduced in Exercise 7.) Elliptic curves over such fields are of major importance in number theory (and cryptography), but we will not discuss these topics further here.

** — 2. Modular functions and modular forms — **

In Exercise 32 of 246A Notes 5, it was shown that two tori and are complex diffeomorphic if and only if one has

for some integers

with

. From this it is not difficult to see that if

are two lattices in

, then

and

are diffeomorphic if and only if

for some

, i.e., the lattices

are complex dilations of each other.

Let us write for the set of all tori quotiented by the equivalence relation of complex diffeomorphism; this is the (classical, level one, noncompactified) modular curve. By the above discussion, this set can also be identified with the set of pairs of linearly independent (over ) complex numbers quotiented by the equivalence relation given implicitly by (13). One can simplify this a little by observing that any pair is equivalent to for some in the upper half-plane , namely either or depending on the relative phases of and ; this quantity is known as the *period ratio*. From (13) (swapping the roles of as necessary), we then see that two pairs are equivalent if one has

for some integers

with

. Recall that the Möbius transformation

preserves

(see Exercise 20 of

246A Notes 5), so the

sign here must actually be positive. We can interpret this in terms of the action of the matrix group

on

by Möbius transformation

and we conclude that two pairs

are equivalent if and only if the period ratios

lie in the same orbit of

. Thus we can identify the modular curve

(as a set, at least) with the quotient space

. Actually if one wished one could replace

here with the projective subgroup

, since the negative identity matrix

acts trivially on

.

If we use the relation to write

we see that

approaches the real line as

if

is non-zero; also, if

is zero, then from

we must have

, and

will either have imaginary part going off to infinity (if

goes to infinity) or real part going to infinity (if

is bounded and

goes to infinity). In all cases we then conclude that

goes to infinity as

goes to infinity, uniformly for

in any fixed compact subset of

, which makes the action of

on

proper (for any compact set

, one has

intersecting

for at most finitely many

. If

acted freely on

(i.e., any element

of

other than the identity and negative identity has no fixed points in

), then the quotient

would be a Riemann surface by the discussion in Section 2 of

246A Notes 5. Unfortunately, this is not quite true. For instance, the point

is fixed by the Möbius transformation

coming from the rotation matrix

of

, and the point

is similarly fixed by the transformation

coming from the matrix

of

. Geometrically, these fixed points come from the fact that the

Gaussian integgers are invariant with respect to rotation by

, while the

Eisenstein integers are invariant with respect to rotation by

. On the other hand, these are basically the only two places where the action is not free:

**Exercise 18** Suppose that is an element of which is fixed by some element of which is not the identity or negative identity. Let be the lattice .
- (i) Show that obeys a dilation invariance for some complex number which is not real.
- (ii) Show that the dilation in part (i) must have magnitude one. (Hint: look at a non-zero element of of minimal magnitude.)
- (iii) Show that there is no rotation invariance with . (Hint: again, work with a non-zero element of of minimal magnitude, and use the fact that is closed under addition and subtraction. It may help to think geometrically and draw plenty of pictures.)
- (iv) Show that is equivalent to either the Gaussian lattice or the Eisenstein lattice , and conclude that the period ratio is equivalent to either or .

**Remark 19** The conformal map on the complex numbers preserves the Gaussian integers and thus descends to a conformal map from the Gaussian torus to itself; similarly the conformal map preserves the Eisenstein integers and thus descends to a conformal map from the Eisenstein torus to itself. These rare examples of complex tori equipped with additional conformal automorphisms are examples of tori (or elliptic curves) endowed with complex multiplication. There are additional examples of elliptic curves endowed with conformal *endomorphisms* that are still considered to have complex multiplication, and have a particularly nice algebraic number theory structure, but we will not pursue this topic further here.

**Remark 20** The fact that the action of on lattices contains fixed points is somewhat annoying, as it prevents one from immediately viewing the modular curve as a Riemann surface. However by passing to a suitable finite index subgroup of , one can remove these fixed points, leading to a theory that is cleaner in some respects. For instance, one can work with the congruence group , which roughly speaking amounts to decorating the lattices (or their tori ) with an additional “-marking” that eliminates the fixed points. This leads to a modification of the theory which is for instance well suited for studying theta functions; the role of the -invariant in the discussion below is then played by the modular lambda function , which also gives a uniformisation of the twice-punctured complex plane . However we will not develop this parallel theory further here.

If we let be the elements of not equivalent to or , and the equivalence class of tori not equivalent to the Gaussian torus or the Eisenstein torus , then can be viewed as the quotient of the Riemann surface by the free and proper action of , so it has the structure of a Riemann surface; can thus be thought of as the Riemann surface with two additional points added. Later on we will also add a third point (known as the *cusp*) to the Riemann surface to compactify it to .

A function on the modular curve can be thought of, equivalently, as a function that is -invariant in the sense that for all and , or equivalently that one has the identity

whenever

and

are integers with

. Similarly if

takes values in the Riemann sphere

rather than

. If

is holomorphic (resp. meromorphic) on

, this will in particular define a holomorphic (resp. meromorphic) function on

, and morally to all of

as well (although we have not yet defined a Riemann structure on all of

).

We define a modular function to be a meromorphic function on that obeys the condition (15), and which also has at most polynomial growth at the cusp in the sense that one has a bound of the form

for all

with sufficiently large imaginary part, and some constants

(this bound is needed for technical reasons to ensure “meromorphic” behaviour at the cusp

, as opposed to an essential singularity). Specialising to the matrices

we see that the condition

(15) in particular implies the

-periodicity

and the inversion law

for all

. Conversely, these two special cases of

(15) imply the general case:

**Exercise 21**

**Exercise 22 (Standard fundamental domain)** Define the *standard fundamental domain* for to be the set
- (i) Show that every lattice is equivalent (up to dilations) to a lattice with , with unique except when it lies on the boundary of , in which case the lack of uniqueness comes either from the pair for some , or from the pair for some . (
*Hint:* arrange so that is a non-zero element of of minimal magnitude.) - (ii) Show that can be identified with the fundamental domain after identifying with for , and with for . Show also that the set is then formed the same way, but first deleting the points from .

We will give some examples of modular functions (beyond the trivial example of constant functions) shortly, but let us first observe that when one differentiates a modular function one gets a more general class of function, known as a modular form. In more detail, observe from (14) that the derivative of the Möbius transformation is , and hence by the chain rule and (15) the derivative of a modular function would obey the variant law

Motivated by this, we can define a (weakly)

*modular form* of weight

for any natural number

to be a meromorphic function

which obeys the modularity relation

for all

and all integers

with

(with the convention that

for any non-zero complex

), and which is meromorphic at the cusp

in the sense of

(16). Thus for instance modular functions are weakly modular forms of weight

. A

*modular form* of weight

is a weakly modular form

of weight

which is holomorphic (not just meromorphic) on

, and also “holomorphic at

” in the sense that

is bounded for

large enough. Note that as viewed a function

of the nome

, a modular form

can be thought of as a certain type of holomorphic function

on the disk

(using the Riemann singularity removal theorem to remove the singularity at the origin

), while weakly modular forms (and in particular modular functions) are certain types of meromorphic functions on this disk. A modular form that vanishes at infinity is known as a

*cusp form*.

**Exercise 23** Let be a natural number. Show that a function obeys (20) if and only if it is -periodic in the sense of (18) and obeys the law
for all .

**Exercise 24 (Lattice interpretation of modular forms)** Let be a modular form of weight . Show that there is a unique function from lattices to complex numbers such that
for all , and such that one has the homogeneity relation
for any lattice and non-zero complex number .

Observe that the product of a modular form of weight and a modular form of weight is a modular form of weight , and that the ratio of two modular forms of weight will be a modular function (if the denominator is not identically zero). Also, the space of modular forms of a given weight is a vector space, as is the space of modular functions. This suggests a way to generate non-trivial modular functions, by first locating some modular forms and then taking suitable rational combinations of these forms.

Somewhat analogously to how we used Lemma 3 to investigate the spaces for divisors on a torus, we will investigate the space of modular forms via the following basic formula:

**Theorem 25 (Valence formula)** Let be a modular form of weight , not identically zero. Then we have
where is the order of vanishing of at , is the order of vanishing of (i.e., viewed as a function of the nome ) at , and ranges over the zeroes of that are not equivalent to , with just one zero counted per equivalence class. (This will be a finite sum.)

Informally, this formula asserts that the point only “deserves” to be counted in with multiplicity due to its order stabiliser, while the point only “deserves” to be counted in with multiplicity due to its order stabiliser. (The cusp has an infinite stabiliser, but this is compensated for by taking the order with respect to the nome variable rather than the period ratio variable .) The general philosophy of weighting points by the reciprocal of the order of their stabiliser occurs throughout mathematics; see this blog post for more discussion.

*Proof:* Firstly, from Exercise 22, we can place all the zeroes in the fundamental domain . When parameterised in terms of the nome , this domain is compact, hence has only finitely many zeros, so the sum in (22) is finite.

As in the proof of Lemma 3(ii), we use the residue theorem. For simplicity, let us first suppose that there are no zeroes on the boundary of the fundamental domain except possibly at the cusp . Then for large enough, we have from the residue theorem that

where

is the closed contour consisting of the polygonal path

concatenated with the circular arc

. From the

-periodicity, the contribution of the two vertical edges

and

cancel each other out. The contribution of the horizontal edge

can be written using the change of variables

as

which by the residue theorem is equal to

. Finally, using the modularity

(21), one calculates that the contribution of the left arc

is equal to

minus the contribution of the right arc

. This gives the proof of the valence theorem in the case that there are no zeroes on the boundary of

.

Suppose now that there is a zero on the right edge of , and hence also on the left edge by periodicity, for some . One can account for this zero by perturbing the contour to make a little detour to the right of (e.g., by a circular arc), and a matching detour to the right of . One can then verify that the same argument as before continues to work, with this boundary zero being counted exactly once. Similarly, if there is a zero on the left arc for some , and hence also at by modularity, one can make a detour slightly above and slightly below (with the two detours being related by the transform to ensure cancellation), and again we can argue as before. If instead there is a zero at , one makes an (approximately) semicircular detour above ; in this case the detour does not cancel out, but instead contributes a factor of in the limit as the radius of the detour goes to zero. Finally, if there is a zero at (and hence also at ), one makes detours by two arcs of angle approximately at these two points; these two (approximate) sixth-circles end up contributing a factor of in the limit, giving the claim.

**Exercise 26 (Quick applications of the valence formula)**
- (i) Let be a modular form of weight , not identically zero. Show that is equal to or an even number that is at least .
- (ii) (Liouville theorem for ) If is a modular form of weight zero, show that it is constant. (Hint: apply the valence theorem to various shifts of by constant.)
- (iii) For , show that the vector space of modular forms of weight is at most one dimensional. (Hint: in these cases, there are a very limited number of solutions to the equation with natural numbers.)
- (iv) Show that there are no cusp forms of weight when or , and for the space of cusp forms of weight is at most one dimensional.
- (v) Show that for any , the space of cusp forms of weight is a subspace of the space of modular forms of weight of codimension at most one, and that both spaces are finite-dimensional.

A basic example of modular forms are provided by the Eisenstein series

that we have already encountered for even integers

greater than two (we ignore the odd Eisenstein series as they vanish). We can view this as a function on

by the formula

Observe that if

are integers with

, then

using the matrix inverse in

. Inserting this into

(23),

(24) we conclude that

(Compare also with Exercise

24.) Also, from

(23),

(24) we have

The series here is locally uniformly convergent for

, so

is holomorphic. Also, using the bounds

for non-zero

, while

where

is the famous

Riemann zeta function
we conclude on summing in

and using the hypothesis

that

In particular,

is bounded at infinity. Summarising, we have established that the Eisenstein series

is a modular form of weight

, which is not identically zero (since it approaches the non-zero value

at the cusp

). Combining this with Exercise

26(iii), we see that we have completely classified the modular forms of weight

for

, namely they are the scalar multiples of

. For instance, the coefficients

and

appearing in the previous section are modular forms of weight

and weight

respectively, and the modular discriminant

from Exercise

10 is a modular form of weight

. From that exercise, this modular form never vanishes on

, hence by the valence formula it must have a simple zero at

, and in particular is a cusp form. From Exercise

26 it is the unique cusp form of weight

, up to constants.

**Exercise 27** Give an alternate proof that is a cusp form, not using the valence identity, by first establishing that and .

We can now create our first non-trivial modular function, the -invariant

The factor of

is traditional, as it gives a nice normalisation at

, as we shall see later. One can take advantage of complex multiplication to compute two special values immediately:

**Lemma 28** We have and .

*Proof:* Using the rotation symmetry we see that , hence which implies that and hence . Similarly, using the rotation symmetry we have , hence . (One can also use the valence formulae to get the vanishing ).

Being modular, we can think of as a map from to . We have the following fundamental fact:

**Proposition 29** The map is a bijection.

*Proof:* Note that for any , if and only if is a zero of . It thus suffices to show that for every , the zeroes of the function in consist of precisely one orbit of . This function is a modular form of weight that does not vanish at infinity (since does not vanish while does). By the valence formula, we thus have

As the orders are all natural numbers, some case checking reveals that there are now only three possibilities:
- has a simple zero at precisely one -orbit, not equivalent to or .
- has a double zero at (and equivalent points), and no other zeroes.
- has a triple zero at (and equivalent points), and no other zeroes.

In any of these three cases, the claim follows.
Note that this proof also shows that has a double zero at and has a triple zero at , but that has a simple zero for any not equivalent to or .

We can now give the entire modular curve the structure of a Riemann surface by declaring to be the coordinate function. This is compatible with the existing Riemann surface structure on since was already holomorphic on this portion of the curve. Any modular function can then factor as for some meromorphic function that is initially defined on the punctured complex plane ; but from meromorphicity of on and at infinity we see that blows up at an at most polynomial rate as one approaches , , or , and so is in fact a meromorphic function on the entire Riemann sphere and is thus a rational function (Exercise 19 of 246A Notes 4). We conclude

**Proposition 30** Every modular function is a rational function of the -invariant .

Conversely, it is clear that every rational function of is modular, thus giving a satisfactory description of the modular functions.

**Exercise 31** Show that every modular function is the ratio of two modular forms of equal weight (with the denominator not identically zero).

**Exercise 32 (All elliptic curves are tori)** Let be two complex numbers with . Show that there is a lattice such that and , so in particular the elliptic curve
is complex diffeomorphic to a torus .

**Remark 33** By applying some elementary algebraic geometry transformations one can show that any (smooth, irreducible) cubic plane curve generated by a polynomial of degree is a Riemann surface complex diffeomorphic to a torus after adding some finite number of points at infinity; also, some degree curves such as
can also be placed in this form. However we will not detail the required transformations here.

A famous application of the theory of the -invariant is to give a short Riemann surface-based proof of the the little Picard theorem (first proven in Theorem 55 of 246A Notes 4):

**Theorem 34 (Little Picard theorem)** Let be entire and non-constant. Then omits at most one point of .

*Proof:* Suppose for contradiction that omits at least two points of . By applying a linear transformation, we may assume that omits the points and . Then is a holomorphic function from to . Since the domain is simply connected, lifts to a holomorphic function from to . Since is complex diffeomorphic to a disk, this lift must be constant by Liouville’s theorem, hence is constant as required. (This is essentially Picard’s original proof of this theorem.)

The great Picard theorem can also be proven by a more sophisticated version of these methods, but it requires some study of the possible behavior of elements of ; see Exercise 37 below.

All modular forms are -periodic, and hence by Proposition 1 should have a Fourier expansion, which is also a Laurent expansion in the nome. As it turns out, the Fourier coefficients often have a highly number-theoretic interpretation. This can be illustrated with the Eisenstein series ; here we follow the treatment in Stein-Shakarchi. To compute the Fourier coefficients we first need a computation:

**Exercise 35** Let and , and let be the nome. Establish the identity
in two different ways:
- (i) By applying the Poisson summation formula (Proposition 3(v) of Notes 2).
- (ii) By first establishing the identity
by applying Proposition 1 to the difference of the two sides, and differentiating in . (It is also possible to establish (27) from differentiating and then manipulating the identities in Exercises 25 or 27 of Notes 1.)

From (25), (26) (and symmetry) one has

and hence by the above exercise

Since

it is not difficult to show that the double sum here is absolutey convergent and can be rearranged as we please. If we group the terms based on the product

we thus have the Fourier expansion

where the

divisor function is defined by

where the sum is over those natural numbers

that divide

. Thus for instance

and

so after some calculation

and therefore

thus the factor of

in the definition of the

-invariant normalises the “residue” of

at infinity to equal

.

**Remark 36** If one expands out a few more terms in the above expansions, one can calculate
The various coefficients in here have several remarkable algebraic properties. For instance, applying this expansion at for a natural number , so that , one obtains the approximation
Now for certain values of , most famously , the torus admits a complex multiplication that allows for computation of the -invariant by algebraic means (think of this as a more advanced version of Lemma 28; it is closely related to the fact that the ring of algebraic integers in admit unique factorisation, see these previous notes for some related discussion). For instance, one can eventually establish that
which eventually leads to the famous approximation
(first observed by Hermite, but also attributed to Ramanujan via an April Fools’ joke of Martin Gardner) which is accurate to twelve decimal places. The remaining coefficients have a remarkable interpretation as dimensions of components of a certain representation of the monster group known as the moonshine module, a phenomenon known as monstrous moonshine. For instance, the smallest irreducible representation of the monster group has dimension , precisely one less than the coefficient of . The Fourier coefficients of the (normalised) modular discriminant,
form a sequence known as the Ramanujan function and obeys many remarkable properties. For instance, there is the totally non-obvious fact that this function is multiplicative in the sense that whenever are coprime; see Exercise 43.

**Exercise 37 (Great Picard theorem)**
- (i) Show that every fractional linear transformation on with , is either of finite order (elliptic case), conjugate to a translation for some after conjugating by another fractional linear transformation (parabolic case), or conjugate to a dilation for some after conjugating by another fractional linear transformation (hyperbolic case). (
*Hint:* study the eigenvalues and eigenvectors of , based on the value of the trace and in particular whether the magnitude of the trace is less than two, equal to two, or greater than two. Note that the trace also has to be an integer.) - (ii) Let be holomorphic. Show that there exists a holomorphic function such that for all , as well as a fractional linear transformation with and such that for all .
- (iii) If the transformation in (ii) is in the elliptic case of (i), show that is bounded in a neighbourhood of , and hence has a removable singularity at the origin. (
*Hint:* will have some finite period and can thus be studied using Proposition 1 after applying a Möbius transform to map to a disk.) - (iv) If the transformation in (ii) is in the hyperbolic case of (i), show that is bounded in a neighbourhood of , and hence has a removable singularity at the origin. (
*Hint:* The standard branch of maps to an annulus, and is invariant with respect to the dilation action . Use this to create a bounded -periodic holomorphic function on .) - (v) If the transformation in (ii) is in the parabolic case of (i), show that exhibits at most polynomial growth as one approaches , and hence has at most a pole at the origin. (
*Hint:* If for instance , then is -periodic and takes values in , and one can now repeat the arguments of (iii). Also use the expansion (28).) - (vi) Use the previous parts of this exercise to give another proof of the great Picard theorem (Theorem 56 of 245A Notes 4): the image of a holomorphic function in a punctured disk with an essential singularity at omits at most one value of .

**Exercise 38 (Dimension of space of modular forms)**
- (i) If is an even natural number, show that the dimension of the space of modular forms of weight is equal to except when is equal to mod , in which case it is equal to . (
*Hint:* for this follows from Exercise 26; to cover the larger ranges of , use the modular discriminant to show that the space of cusp forms of weight is isomorphic to the space of modular forms of weight . - (ii) If is an even natural number, show that a basis for the space of modular forms of weight is provided by the powers where range over natural numbers (including zero) with .

Thus far we have constructed modular forms and modular functions starting from Eisenstein series . There is another important, and seemingly quite different, way to generate modular forms coming from theta functions. Typically these functions are not *quite* modular in the sense given in these notes, but are close enough that after some manipulation one can transform theta functions into modular forms. The simplest example of a theta function is the Jacobi theta function

which is easily seen to be a holomorphic function on

that goes to zero as

. It is not quite a modular form, but is

-periodic (instead of

-periodic) in the sense that

for all

, and from Poisson summation (see Exercise 7 of

Notes 2) we have the variant

of the modularity relation

(21), using the standard branch of the square root. This is not quite modular in nature, but a slight variant of the theta function fares better:

**Exercise 39** Define the Dedekind eta function by the formula
or in terms of the nome
where is one of the roots of .

**Remark 40** The relationship between and the power of the eta function can be interpreted (after some additional effort) as a relation between the modular discriminant and the theta function of a certain highly symmetric -dimensional lattice known as the Leech lattice, but we will not pursue this connection further here.

The function has a remarkable factorisation coming from Euler’s pentagonal number theorem

so that

There are many proofs of the pentagonal number theorem in the literature. One approach is to first establish the more general

Jacobi triple product identity:

**Theorem 41 (Jacobi triple product identity)** For any and , one has

Observe that by replacing by and with we have

and this gives the identity

(31) after splitting the integers into the three residue classes

modulo

. One can obtain many further identities of this type by other substitutions; for instance, by setting

in the triple product identity, one obtains

*Proof:* Let us denote the left-hand side and right-hand side of (33) by and respectively. For fixed , both sides are clearly holomorphic in , with . Our strategy in showing that and agree (following Stein-Shakarchi) is to first observe that they have many of the same periodicity properties. We clearly have -periodicity

From the identity

we also have the modified

-periodicity

similarly from the telescoping products

and

we conclude that

also obeys the same modified

-periodicity

Thus the ratio

is meromorphic and doubly periodic. Furthermore, one checks that

only vanishes when

is equal to

modulo

with a simple zero at those locations, so the ratio

has at most a single simple pole on the torus

and thus constant by the discussion after Lemma

6 (alternatively, one can show that

also vanishes at this point and apply Proposition

2). Thus there is some quantity

depending only on

for which we have the identity

for all

. To exploit this, we first set

and replace

by

to conclude that

but on rearranging the absolutely convergent product we have

and thus

If instead we set

and not modify

, we have

the contribution of

and

on the left-hand sides cancel when

is odd, so that on making the substitution

we obtain

while from rearranging an absolutely convergent product we have

and thus by difference of two squares

Comparing this with

(34) we obtain the surprising additional symmetry

On the other hand, taking limits in (say)

(35) we see that

as

. If we then iterate

(36) we conclude that

and the claim follows.

**Remark 42** Another equivalent form of (32) is
where is the partition function of – the number of ways to represent as the sum of positive integers (up to rearrangement). Among other things, this formula can be used to ascertain the asymptotic growth of (which turns out to roughly be of the order of , as famously established by Hardy and Ramanujan).

Theta functions can be used to encode various number-theoretic quantities involving quadratic forms, such as sums of squares. For instance, from (30) and collecting terms one obtains the formula

for any natural number

, where

denotes the number of ways to express a natural number

as the sum

of

squares of integers. From Fourier inversion (Proposition

1 and a rescaling) one then has a representation

for any

, which allows one to obtain asymptotics for

when

is large through estimation of the theta function (this is an example of the circle method); moreover, explicit identities relating the theta function to other near-modular forms (such as the Eisenstein series and their relatives) can be used to obtain exact formulae for

for small values of

that can be used for instance to establish the famous

Lagrange four-square theorem that all natural numbers are the sum of four squares. We refer the reader to the Stein-Shakarchi text for an exposition of this connection.

**Exercise 43 (Hecke operators)** Let be a natural number.
Simultaneous eigenfunctions of the Hecke operators are known as *Hecke eigenfunctions* and are of major importance in number theory.