Chapter 5

Let \(\mathcal{F}\) be a fundamental parallelogram of the lattice \(L\). Show $$ \mathbb{C}=\bigcup_{\omega \in L}(\omega+\mathcal{F}) $$

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 \(\sigma(z)=\sigma(z ; L)\) be the WEIERSTRASS \(\sigma\)-function for the lattice \(L=\mathbb{Z} \omega_{1}+\) \(\mathbb{Z} \omega_{2} .\) The function $$ \zeta(z):=\zeta(z ; L):=\frac{\sigma^{\prime}(z)}{\sigma(z)} $$ is called the Weierstrass \(\zeta\)-function for the lattice \(L .\) (This function should not be confused with the RIEMANN \(\zeta\)-function!) Then \(-\zeta^{\prime}(z)=\wp(z)\) is the WeIERSTRASS \(\wp\)-function for the lattice \(L\). Assume \(\operatorname{Im}\left(\omega_{2} / \omega_{1}\right)>0\) Show: Using the notation \(\eta_{\nu}:=\zeta\left(z+\omega_{\nu}\right)-\zeta(z)\) for \(\nu=1,2\), the following relation of LEGENDRE is true: LEGENDRE's Relation, $$ \eta_{1} \omega_{2}-\eta_{2} \omega_{1}=2 \pi \mathrm{i} $$

Prove the structure theorem for discrete subgroups \(L \subset \mathbb{C}\). Hint. If \(L \neq\\{0\\}\), then there exists a period \(\omega_{1} \neq 0\) in \(L\) of minimal absolute value. Then $$ L \cap \mathbb{R} \omega_{1}=\mathbb{Z} \omega_{1} $$ 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.

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 \(\left(\frac{9}{4},-\frac{21}{4}\right)\) Hint. Find the (third) intersection point of the elliptic curve with its tangent in \((2,4)\)

Prove, using the Doubling Formula of the WEIERSTRASS \(\wp\)-function, the FAGNANO Doubling Formula for the lemniscate arcs, $$ 2 \int_{0}^{x} \frac{1}{\sqrt{1-t^{4}}} d t=\int_{0}^{y} \frac{1}{\sqrt{1-t^{4}}} d t \quad \text { with } \quad y=2 x \frac{\sqrt{1-x^{4}}}{1+x^{4}} $$

The number of minimal vectors (i.e. non-zero vectors of minimal modulus) in a lattice \(L\) is 2,4 or 6 . Give also explicit examples for each case.

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 domain, which satisfies the same algebraic differential equation as \(\wp\), i.e. $$ f^{\prime 2}=4 f^{3}-g_{2} f-g_{3} $$ Show that \(f\) is the composition of \(\wp\) with a translation, i.e. there exists an \(a \in \mathbb{C}\) with \(f(z)=\wp(z+a)\).

For \(z, a \in \mathbb{C} \backslash L\) we have the relations $$ \wp(z)-\wp(a)=-\frac{\sigma(z+a) \sigma(z-a)}{\sigma(z)^{2} \sigma(a)^{2}} $$ and $$ \wp^{\prime}(a)=-\frac{\sigma(2 a)}{\sigma(a)^{4}} $$

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 functions with prescribed principal parts Let \(f\) be an elliptic function for the lattice \(L\). 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 ensures the relation $$ \sum_{j=1}^{n} a_{1, j}=0 $$ Show: (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 by means 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 such that \(\sum a_{1, j}=0\) 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 $$ \sum_{\nu=1}^{l_{j}} \frac{a_{\nu, j}}{\left(z-b_{j}\right)^{\nu}} $$

Two lattices \(L=\mathbb{Z} \omega_{1}+\mathbb{Z} \omega_{2}\) and \(L^{\prime}=\mathbb{Z} \omega_{1}^{\prime}+\mathbb{Z} \omega_{2}^{\prime}\) coincide iff there exists a matrix with integral entries \(\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)\) and determinant \(\pm 1\) with the property $$ \left(\begin{array}{l} \omega_{1}^{\prime} \\ \omega_{2}^{\prime} \end{array}\right)=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)\left(\begin{array}{l} \omega_{1} \\ \omega_{2} \end{array}\right) $$

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 \(\omega_{1}, \omega_{2}\). Show: The EUCLIDian volume of the fundamental parallelogram is \(\left|\operatorname{Im}\left(\bar{\omega}_{1} \omega_{2}\right)\right|\). This formula is independent of the choice of the basis.

Any elliptic function of order \(\leq 2\) with period lattice \(L\), whose pole set is contained in \(L\), is of the form \(z \rightarrow a+b \wp(z)\).

We call a meromorphic function \(f: \mathbb{C} \rightarrow \overline{\mathbb{C}}\) "real", iff \(f(\bar{z})=\overline{f(z)}\) for all \(z \in \mathbb{C}\). A lattice \(L \subset \mathbb{C}\) is called "real", iff \(\omega \in L\) implies \(\bar{\omega} \in L\) (i.e. iff \(L\) is invariant under complex conjugation). Show tha the following properties are equivalent. (a) \(g_{2}(L), g_{3}(L) \in \mathbb{R}\). (b) \(G_{n} \in \mathbb{R}\) for all (even) \(n \geq 4\). (c) The \(\wp\)-function is real. (d) The lattice \(L\) is real.

We are interested in alternating \(\mathbb{R}\)-bilinear maps (forms) $$ A: \mathbb{C} \times \mathbb{C} \longrightarrow \mathbb{R} $$ Show: (a) Any such map \(A\) is of the form $$ A(z, w)=h \operatorname{Im}(z \bar{w}) $$ with a uniquely determined real number \(h\). We have explicitly \(h=A(1, i)\). (b) Let \(L \subset \mathbb{C}\) be a lattice. Then \(A\) is called a Riemannian form with respect to \(L\), iff \(h\) is positive, and \(A\) only takes integral values on \(L \times L\). If $$ L=\mathbb{Z} \omega_{1}+\mathbb{Z} \omega_{2}, \quad \operatorname{Im} \frac{\omega_{2}}{\omega_{1}}>0 $$ then the formula $$ A\left(t_{1} \omega_{1}+t_{2} \omega_{2}, s_{1} \omega_{1}+s_{2} \omega_{2}\right):=\operatorname{det}\left(\begin{array}{ll} t_{1} & s_{1} \\ t_{2} & s_{2} \end{array}\right) $$ defines a RIEMANNian form \(A\) on \(L\). c) A non-constant analytic function \(\Theta: \mathbb{C} \rightarrow \mathbb{C}\) is called a theta function for the lattice \(L \subset \mathbb{C}\), iff it satisfies an equation of the type $$ \Theta(z+\omega)=e^{a_{\omega} z+b_{\omega}} \cdot \Theta(z) $$ for all \(z \in \mathbb{C}\), and all \(\omega \in L .\) Here, \(a_{\omega}\) and \(b_{\omega}\) are constants that may depend on \(\omega\), but not on \(z\). The WEIERSTRASS \(\sigma\)-function for the lattice \(L\) is in this sense a theta function. Show the existence of a RIEMANNian form \(A\) with respect to \(L\), such that $$ A(\omega, \lambda)=\frac{1}{2 \pi \mathrm{i}}\left(a_{\omega} \lambda-\omega a_{\lambda}\right) \text { for all } \omega, \lambda \in L $$

The group \(\mathbb{Z}+\mathbb{Z} \sqrt{2}\) is dense in \(\mathbb{R}\).

A lattice is called rectangular, iff it admits a basis \(\omega_{1}, \omega_{2}\), such that \(\omega_{1}\) is real and \(\omega_{2}\) is purely imaginary. A lattice \(L\) is called rhombic, iff it admits a basis \(\omega_{1}, \omega_{2}\), such that \(\omega_{2}=\bar{\omega}_{1}\) Show that a lattice is real, iff it is either rectangular or rhombic.

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 \(P_{\omega}\) with the property $$ f(z+\omega)=f(z)+P_{\omega}(z) $$ Then \(f\) is a polynomial.

The WEIERSTRASS \(\wp\)-function with respect to a rectangular lattice \(L=\mathbb{Z} \omega_{1}+\) \(\mathbb{Z} \omega_{2}\), where \(\omega_{1} \in \mathbb{R}_{+}^{\bullet}\) and \(\omega_{2} \in \mathrm{i} \mathbb{R}_{+}^{\bullet}\), takes real values on the boundary and on the middle lines of the fundamental rectangle.

Another variant of the First LIOUVILLE Theorem: Let \(f\) be an entire function, and let \(L\) be a lattice in \(\mathbb{C}\). For any lattice point \(\omega \in L\) let there exist a number \(C_{\omega} \in \mathbb{C}\) with the property $$ f(z+\omega)=C_{\omega} f(z) $$ Then $$ f(z)=C e^{a z} $$ for suitable constants \(C\) and \(a\). Hint. Without loss of generality, we can assume \(\omega_{1}=1\) and \(C_{\omega_{1}}=1\). Use the FOURIER series of \(f\). Another proof can be given by showing that \(f^{\prime} / f\) is constant.

In this exercise we use the notions "extension of fields" \(k \subset K\) 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 elementary facts from algebra: Let us assume, 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. Then any \(n+1\) elements of \(K\) are algebraically dependent over \(k\). Show that any two elliptic functions (for the same lattice \(L\) ) are algebraically dependent over \(\mathbb{C}\).

Let \(L\) be a lattice. Show that there does not exist any elliptic function \(f \in K(L)\), with \(\mathbb{C}(f)=K(L)\), i.e. such that any other elliptic function can be written as a rational function in \(f\) Hint. Analyze the equation \(f(z)=f(w)\), and show that if there would exist an \(f\) with the above property then \(f\) would be an elliptic function of order 1 .