Chapter 5

If \(L \subset \mathbb{C}\) is a lattice, then the formula $$ \sum_{\omega \in L} \frac{1}{(z-\omega)^{n}} $$ defines for any \(n \geq 3\) an elliptic function of order \(n\). Which is the connection with the WEIERSTRASS \(p\)-function?

The zeros \(e_{1}, e_{2}\) and \(e_{3}\) of the polynomial \(4 X^{3}-g_{2} X-g_{3}\) are all real, iff \(g_{2}, g_{3}\) are real, and the discriminant \(\Delta=g_{2}^{3}-27 g_{3}^{2}\) is non-negative.

Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be a non-constant meromorphic function. The set of its periods $$ L_{f}:=\\{\omega \in \mathbb{C} ; \quad f(z+\omega)=f(z) \text { for all } z \in \mathbb{C}\\} $$ is a discrete subgroup of \(\mathrm{C}\).

The already introduced instruments are at a hair length enough to manage the following exercise. Let \(L \subset \mathbb{C}\) be a lattice, and let \(P(t)=4 t^{3}-g_{2} t-g_{3}\) be the associated cubic polynomial. Let \(\alpha:[0,1] \rightarrow \mathbb{C}\) be a closed path, which avoids the zeros of the polynomial. Finally, let \(h:[0,1] \rightarrow \mathbb{C}\) be a continuous function with the properties $$ h(t)^{2}=\frac{1}{P(\alpha(t))} \quad \text { and } \quad h(0)=h(1) $$ The number $$ \int_{0}^{1} h(t) \alpha^{\prime}(t) d t=\int_{0}^{1} \frac{\alpha^{\prime}(t)}{\sqrt{P(\alpha(t))}} d t $$ is called a period of the elliptic integral \(\int 1 / \sqrt{P(z)} d z .\) Show that the periods of the elliptic integral lie in \(L\). (One can supplementary show, that \(L\) is precisely the set of all periods of the elliptic integral.)This fact opens a new approach to the problem, how to realize each pair of complex numbers \(\left(g_{2}, g_{3}\right)\) with non-vanishing discriminant as a pair \(\left(g_{2}(L), g_{3}(L)\right)\) with a suitable lattice \(L\). This parallel access will be taken up in the next book, in connection with the theory of RIEMANN surfaces. In this book, we are arguing differently (V.8.9). A detailed analysis delivers in concrete situations explicit formulas for a basis of \(L\) : Assume the zeros \(e_{1}, e_{2}\) and \(e_{3}\) of \(4 X^{3}-g_{2} X-g_{3}\) are all real, pairwise different, and indexed to satisfy \(e_{2}>e_{3}>e_{1}\) Then the integrals $$ \omega_{1}=2 \mathrm{i} \int_{-\infty}^{c_{1}} \frac{1}{\sqrt{-4 t^{3}+g_{2} t+g_{3}}} d t \quad \text { and } \quad \omega_{2}=2 \int_{e_{2}}^{\infty} \frac{1}{\sqrt{4 t^{3}-g_{2} t-g_{3}}} d t $$ are a basis of the lattice \(L\).

The surjectivity of \(j: \mathbb{H} \longrightarrow \mathbb{C}\) was motivated as follows: (a) \(j(\mathbb{H})\) is by the Open Mapping Theorem open in \(\mathbb{C}\) and non-empty. (b) \(j(\mathbb{H})\) is closed in \(\mathbb{C}\). This implies \(j(\mathbb{H})=\mathbb{C}\), since \(\mathbb{C}\) is connected. Fill in the details.

Prove the structure theorem for discrete subgroups \(L \subset \mathrm{C} .\) Hint. If \(L \neq\\{0\\}\), then there exists a period \(\omega_{1} \neq 0\) in \(L\) of minimal absolute value. Then $$ L \cap \mathrm{R} \omega_{1}=Z \omega_{1} $$ If \(L\) lies in the real line \(\mathbb{R} \omega_{1}\) generated by \(\omega_{1}\), then the structure theorem easily follows. Else, there exists an \(\omega_{2}\) in \(L\), which does not lie in \(\mathbb{R} \omega_{1}\), having minimal absolute value with this property. Show then \(L=\mathbb{Z} \omega_{1}+\mathbb{Z} \omega_{2}\) From the structure theorem we can prove: If \(L \subset \mathbb{C}\) is a discrete subgroup which contains a lattice, then it is itself a lattice. In particular, any group \(L^{\prime}\) which sits between two lattices \(L\) and \(L^{\prime \prime}\). \(L \subset L^{\prime} \subset L^{\prime \prime}\), is also a lattice.

For an odd elliptic function associated to the lattice \(L\) the half-lattice points \(\omega / 2, \omega \in L\), are either zeros or poles.

Let \(L \subset \mathbb{C}\) be a lattice with the property \(g_{2}(L)=8\) and \(g_{3}(L)=0 .\) The point \((2,4)\) lies on the affine elliptic curve \(y^{2}=4 x^{3}-8 x\). Let \(+\) be the addition (for points on the corresponding projective curve). Show that \(2 \cdot(2,4):=(2,4)+\) \((2,4)\) is the point \(\left(\frac{9}{4},-\frac{21}{4}\right)\).

Let \(f\) be an elliptic function of order \(m\). Then its derivative \(f^{\prime}\) is also an elliptic function of some order \(n\), and the following double inequality holds: $$ m+1 \leq n \leq 2 m $$ Construct examples for the extreme cases \(n=m+1\) and \(n=2 m\).

Let us set \(g_{2}=g_{2}(L), g_{3}=g_{3}(L)\), for the \(g\)-invariants of a fixed lattice \(L .\) Let \(f\) be a meromorphic, non-constant function in some non- empty domain, which satisfies the same algebraic differential equation as \(\wp=\wp(\cdot, L)\), i.e. $$ f^{r 2}=4 f^{3}-g_{2} f-g_{3} $$ Show that \(f\) is the composition of \(\wp\) with a translation, i.e. there exists an \(a \in \mathbb{C}\) with \(f(z)=\rho(z+a)\) for any \(z \in C\) Hint. Consider a local inverse function \(f^{-1}\) of \(f\), and reformulate the hypothesis and conclusion for the auxiliary function \(h:=f^{-1} \circ \rho\).

Let \(f\) and \(g\) be elliptic functions for the same lattice. (a) If \(f\) and \(g\) have the same poles, and for each pole respectively the same principal parts, then \(f\) and \(g\) differ by an additive constant. (b) If \(f\) and \(g\) have the same pole set and the same zero set, and if for any pole or zero the corresponding multiplicities coincide, then \(f\) and \(g\) differ by a multiplicative constant.

Construction of elliptic function with prescribed principal parts Let \(f\) be an elliptic function for the \(L\). We choose \(b_{1}, \ldots, b_{n}\) to be a system of representatives modulo \(L\) for the poles of \(f\), and we consider for each \(j\) the principal part of \(f\) in the pole \(b_{j}:\) $$ \sum_{\nu=1}^{l_{j}} \frac{a_{\nu, j}}{\left(z-b_{j}\right)^{\nu}} $$ The Second LIOUVILLE Theorem insures the relation $$ \sum_{j=1}^{n} a_{1, j}=0 $$(a) Let \(c_{1}, \ldots, c_{n} \in \mathbb{C}\) be given numbers, and let \(b_{1}, \ldots, b_{n}\) modulo \(L\) be a set of different points in \(\mathbb{C} / L\). The function $$ h(z):=\sum_{j=1}^{n} c_{j} \zeta\left(z-b_{j}\right) $$ constructed with the help of the WeIERSTRASS \(\zeta\)-function, is then elliptic, iff \(\sum_{j=1}^{n} c_{j}=0\) (b) Let \(b_{1}, \ldots, b_{n}\) be pairwise different modulo \(L\), and let \(l_{1}, \ldots, l_{n}\) be prescribed natural numbers. Let \(a_{\nu, j}\left(1 \leq j \leq n, 1 \leq \nu \leq l_{j}\right)\) be complex numbers with \(\sum a_{1, j}=0\) and \(a_{l_{j}, j} \neq 0\) for all \(j\) Then there exists an elliptic function for the lattice \(L\), having poles modulo \(L\) exactly in the points \(b_{1}, \ldots, b_{n}\), and having the corresponding principal parts respectively equal to $$ \sum_{\nu=1}^{l_{j}} \frac{a_{\nu, j}}{\left(z-b_{j}\right)^{\nu}} $$

Let \(L \subset \mathbb{C}\) be a lattice, and let \(b_{1}, b_{2} \in \mathbb{C}\) with \(b_{1}-b_{2} \notin L .\) Find an elliptic function for the lattice \(L\), having its poles exactly in \(b_{1}\) and \(b_{2}\), and having the corresponding principal parts $$ \frac{1}{z-b_{1}}+\frac{2}{\left(z-b_{1}\right)^{2}} \quad \text { and } \quad \frac{-1}{z-b_{2}} $$

Let $$ \mathcal{F}:=\left\\{z \in \mathbb{C} ; \quad z=t_{1} \omega_{1}+t_{2} \omega_{2}, 0 \leq t_{1}, t_{2} \leq 1\right\\} $$ be the fundamental region of the lattice \(L=\mathbb{Z} \omega_{1}+\mathbb{Z} \omega_{2}\) with respect to a fixed basis \(\left(\omega_{1}, \omega_{2}\right)\) Show: The EucLiDian volume of the fundamental parallelogram is \(\left|\operatorname{Im}\left(\bar{\omega}_{1} \omega_{2}\right)\right|\). This formula in independent of the choice of the basis.

We call a meromorphic function \(f: \mathrm{C} \rightarrow \overline{\mathbb{C}}\) "real", iff \(f(\bar{z})=\overline{f(z)}\) for all \(z \in \mathbb{C}\). (If \(z\) is a pole, we formally set \(\bar{\infty}=\infty\), to obtain the conjugation map \(\overline{\mathbb{C}} \rightarrow \overline{\mathbb{C}}\) fitting with \(\left.\mathbb{P}^{1}(\mathrm{C}) \rightarrow \mathbb{P}^{1}(\mathbb{C}),\left[z_{0}, z_{1}\right] \rightarrow\left[\bar{z}_{0}, \bar{z}_{1}\right] .\right)\) A lattice \(L \subset \mathbb{C}\) is called "real", iff \(\omega \in L\) implies \(\bar{\omega} \in L\) (i.e. iff \(L\) is invariated by the complex conjugation as a set). Show the equivalence of the following propositions: (a) \(g_{2}(L), g_{3}(L) \in \mathbb{R}\). (b) \(G_{n} \in \mathbb{R}\) for all (even) \(n \geq 4\). (c) The \(\varphi\)-function is real. (d) The lattice \(L\) is real.

Prove the following generalization of the First LIOUVILLE Theorem: Let \(f\) be an entire function, and let \(L\) be a lattice in \(\mathbb{C}\). For any lattice point \(\omega \in L\) we assume the existence of a polynomial function \(P_{\omega}\) with the property $$ f(z+\omega)=f(z)+P_{\omega}(z) $$ Then \(f\) is a polynomial function.

In this exercise we use the notions "extension of fields" (for an inclusion) \(k \subset K\) and "algebraic dependence". The elements \(a_{1}, \ldots, a_{n}\) in \(K\) are called algebraically dependent over \(k\), iff there exists a non- zero polynomial \(P\) in \(n\) variables with coefficients in \(k, P \in k\left[X_{1}, \ldots, X_{n}\right]\), such that \(P\left(a_{1}, \ldots, a_{n}\right)=0\). We use the following known fact in GaLois theory: Let us suppose, that there exist \(n\) elements \(a_{1}, \ldots, a_{n} \in K\), such that \(K\) is algebraic over the field \(k\left(a_{1}, \ldots, a_{n}\right)\) generated by these elements over \(k .\) Then any set of \(n+1\) elements of \(K\) are algebraically dependent over \(k\). Show that any two elliptic functions (for the same lattice \(L\) ) are algebraically dependent over \(\mathbb{C}\).