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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 ${\sin(2\pi x)}$, ${\cos(2\pi x)}$, which are ${1}$-periodic in the sense that

$\displaystyle \sin(2\pi(x+1)) = \sin(2\pi x); \quad \cos(2\pi (x+1)) = \cos(2\pi x).$

By taking various polynomial combinations of ${\sin(2\pi x)}$ and ${\cos(2\pi x)}$ we obtain more general trigonometric polynomials that are ${1}$-periodic; and the theory of Fourier series tells us that all other ${1}$-periodic functions (with reasonable integrability conditions) can be approximated in various senses by such polynomial combinations. Using Euler’s identity, one can use ${e^{2\pi ix}}$ and ${e^{-2\pi ix}}$ in place of ${\sin(2\pi x)}$ and ${\cos(2\pi x)}$ 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 ${1}$. ${1}$-periodic functions ${f: {\bf R} \rightarrow {\bf C}}$ can also be identified (by abuse of notation) with functions ${f: {\bf R}/{\bf Z} \rightarrow {\bf C}}$ on the quotient space ${{\bf R}/{\bf Z}}$ (known as the additive ${1}$-torus or additive unit circle), or with functions ${f: [0,1] \rightarrow {\bf C}}$ on the fundamental domain (up to boundary) ${[0,1]}$ of that quotient space with the periodic boundary condition ${f(0)=f(1)}$. The map ${x \mapsto (\cos(2\pi x), \sin(2\pi x))}$ also identifies the additive unit circle ${{\bf R}/{\bf Z}}$ with the geometric unit circle ${S^1 = \{ (x,y) \in {\bf R}^2: x^2+y^2=1\} \subset {\bf R}^2}$, thanks in large part to the fundamental trigonometric identity ${\cos^2 x + \sin^2 x = 1}$; this can also be identified with the multiplicative unit circle ${S^1 = \{ z \in {\bf C}: |z|=1 \}}$. (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 ${x,y}$ of the geometric unit circle, or Laurent polynomials of the complex variable ${z}$.

What about periodic functions on the complex plane? We can start with singly periodic functions ${f: {\bf C} \rightarrow {\bf C}}$ which obey a periodicity relationship ${f(z+\omega)=f(z)}$ for all ${z}$ in the domain and some period ${\omega \in {\bf C} \backslash \{0\}}$; such functions can also be viewed as functions on the “additive cylinder” ${\omega {\bf Z} \backslash {\bf C}}$ (or equivalently ${{\bf C} / \omega {\bf Z}}$). We can rescale ${\omega=1}$ as before. For holomorphic functions, we have the following characterisations:

Proposition 1 (Description of singly periodic holomorphic functions)
In both cases, the coefficients ${a_n}$ can be recovered from ${f}$ by the Fourier inversion formula

$\displaystyle a_n = \int_{\gamma_{z_0 \rightarrow z_0+1}} f(z) e^{-2\pi i nz}\ dz \ \ \ \ \ (5)$

for any ${z_0}$ in ${{\bf C}}$ (in case (i)) or ${{\bf H}}$ (in case (ii)).

Proof: If ${f: {\bf C} \rightarrow {\bf C}}$ is ${1}$-periodic, then it can be expressed as ${f(z) = F(q) = F(e^{2\pi i z})}$ for some function ${F: {\bf C} \backslash \{0\} \rightarrow {\bf C}}$ on the “multiplicative cylinder” ${{\bf C} \backslash \{0\}}$, since the fibres of the map ${z \mapsto e^{2\pi i z}}$ are cosets of the integers ${{\bf Z}}$, on which ${f}$ is constant by hypothesis. As the map ${z \mapsto e^{2\pi i z}}$ is a covering map from ${{\bf C}}$ to ${{\bf C} \backslash \{0\}}$, we see that ${F}$ will be holomorphic if and only if ${f}$ is. Thus ${F}$ must have a Laurent series expansion ${F(q) = \sum_{n=-\infty}^\infty a_n q^n}$ with coefficients ${a_n}$ 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 ${z \mapsto e^{2\pi i z}}$ is also a covering map from ${{\bf H}}$ to the punctured disk ${D(0,1) \backslash \{0\}}$, so we can argue as before except that now ${F}$ is a bounded holomorphic function on the punctured disk. By the Riemann singularity removal theorem (Exercise 35 of 246A Notes 3) ${F}$ extends to be holomorphic on all of ${D(0,1)}$, and thus has a Taylor expansion ${F(q) = \sum_{n=0}^\infty a_n q^n}$ for some coefficients ${a_n}$ obeying (4). The argument now proceeds as with part (i). $\Box$

The additive cylinder ${{\bf Z} \backslash {\bf C}}$ and the multiplicative cylinder ${{\bf C} \backslash \{0\}}$ can both be identified (on the level of smooth manifolds, at least) with the geometric cylinder ${\{ (x,y,z) \in {\bf R}^3: x^2+y^2=1\}}$, but we will not use this identification here.

Now let us turn attention to doubly periodic functions of a complex variable ${z}$, that is to say functions ${f}$ that obey two periodicity relations

$\displaystyle f(z+\omega_1) = f(z); \quad f(z+\omega_2) = f(z)$

for all ${z \in {\bf C}}$ and some periods ${\omega_1,\omega_2 \in {\bf C}}$, which to avoid degeneracies we will assume to be linearly independent over the reals (thus ${\omega_1,\omega_2}$ are non-zero and the ratio ${\omega_2/\omega_1}$ is not real). One can rescale ${\omega_1,\omega_2}$ by a common scaling factor ${\lambda \in {\bf C} \backslash \{0\}}$ to normalise either ${\omega_1=1}$ or ${\omega_2=1}$, 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 ${2}$-torus ${\Lambda \backslash {\bf C}}$, where ${\Lambda}$ is the lattice ${\Lambda := \omega_1 {\bf Z} + \omega_2 {\bf Z}}$, or with functions ${f}$ on the solid parallelogram bounded by the contour ${\gamma_{0 \rightarrow \omega_1 \rightarrow \omega_1+\omega_2 \rightarrow \omega_2 \rightarrow 0}}$ (a fundamental domain up to boundary for that torus), obeying the boundary periodicity conditions

$\displaystyle f(z+\omega_1) = f(z)$

for ${z}$ in the edge ${\gamma_{\omega_2 \rightarrow 0}}$, and

$\displaystyle f(z+\omega_2) = f(z)$

for ${z}$ in the edge ${\gamma_{\omega_0 \rightarrow 1}}$.

Within the world of holomorphic functions, the collection of doubly periodic functions is boring:

Proposition 2 Let ${f: {\bf C} \rightarrow {\bf C}}$ be an entire doubly periodic function (with periods ${\omega_1,\omega_2}$ linearly independent over ${{\bf R}}$). Then ${f}$ is constant.

In the language of Riemann surfaces, this proposition asserts that the torus ${\Lambda \backslash {\bf C}}$ 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 ${\gamma_{0 \rightarrow \omega_1 \rightarrow \omega_1+\omega_2 \rightarrow \omega_2 \rightarrow 0}}$ is compact, hence ${f}$ is bounded on this domain, hence bounded on all of ${{\bf C}}$ by double periodicity. The claim now follows from Liouville’s theorem. (One could alternatively have argued here using the compactness of the torus ${(\omega_1 {\bf Z} + \omega_2 {\bf Z}) \backslash {\bf C}}$. $\Box$

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 ${{\bf C} \cup \{\infty\}}$. As it turns out, a particularly fundamental example of such a function is the Weierstrass elliptic function

$\displaystyle \wp(z) := \frac{1}{z^2} + \sum_{z_0 \in \Lambda \backslash 0} \frac{1}{(z-z_0)^2} - \frac{1}{z_0^2} \ \ \ \ \ (6)$

which plays a role in doubly periodic functions analogous to the role of ${x \mapsto \cos(2\pi x)}$ for ${1}$-periodic real functions. This function will have a double pole at the origin ${0}$, and more generally at all other points on the lattice ${\Lambda}$, but no other poles. The derivative

$\displaystyle \wp'(z) = -2 \sum_{z_0 \in \Lambda} \frac{1}{(z-z_0)^3} \ \ \ \ \ (7)$

of the Weierstrass function is another doubly periodic meromorphic function, now with a triple pole at every point of ${\Lambda}$, and plays a role analogous to ${x \mapsto \sin(2\pi x)}$. Remarkably, all the other doubly periodic meromorphic functions with these periods will turn out to be rational combinations of ${\wp}$ and ${\wp'}$; furthermore, in analogy with the identity ${\cos^2 x+ \sin^2 x = 1}$, one has an identity of the form

$\displaystyle \wp'(z)^2 = 4 \wp(z)^3 - g_2 \wp(z) - g_3 \ \ \ \ \ (8)$

for all ${z \in {\bf C}}$ (avoiding poles) and some complex numbers ${g_2,g_3}$ that depend on the lattice ${\Lambda}$. Indeed, much as the map ${x \mapsto (\cos 2\pi x, \sin 2\pi x)}$ creates a diffeomorphism between the additive unit circle ${{\bf R}/{\bf Z}}$ to the geometric unit circle ${\{ (x,y) \in{\bf R}^2: x^2+y^2=1\}}$, the map ${z \mapsto (\wp(z), \wp'(z))}$ turns out to be a complex diffeomorphism between the torus ${(\omega_1 {\bf Z} + \omega_2 {\bf Z}) \backslash {\bf C}}$ and the elliptic curve

$\displaystyle \{ (z, w) \in {\bf C}^2: z^2 = 4w^3 - g_2 w - g_3 \} \cup \{\infty\}$

with the convention that ${(\wp,\wp')}$ maps the origin ${\omega_1 {\bf Z} + \omega_2 {\bf Z}}$ of the torus to the point ${\infty}$ 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 ${\wp, \wp'}$). 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 ${\Lambda \backslash {\bf C}}$. 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 ${X_0(1)}$. This curve can be described in a number of ways. On the one hand, it can be viewed as the upper half-plane ${{\bf H} = \{ z: \mathrm{Im}(z) > 0 \}}$ quotiented out by the discrete group ${SL_2({\bf Z})}$; on the other hand, by using the ${j}$-invariant, it can be identified with the complex plane ${{\bf C}}$; alternatively, one can compactify the modular curve and identify this compactification with the Riemann sphere ${{\bf C} \cup \{\infty\}}$. (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 ${j}$-invariant) can be viewed as ${SL_2({\bf Z})}$-invariant functions on ${{\bf H}}$, and include the important class of modular functions; they naturally generalise to the larger class of (weakly) modular forms, which are functions on ${{\bf H}}$ which transform in a very specific way under ${SL_2({\bf Z})}$-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 ${\wp}$. More number theoretic examples of modular forms include (suitable powers of) theta functions ${\theta}$, and the modular discriminant ${\Delta}$. Modular forms are ${1}$-periodic functions on the half-plane, and hence by Proposition 1 come with Fourier coefficients ${a_n}$; 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 ${SL_2({\bf Z})}$ (such as congruence groups) and to other domains than the half-plane ${{\bf H}}$, 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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