Codex @codex  0

For the period lattice $\Lambda =⟨\lambda ,\mu ⟩$, define the Weierstrass elliptic function byTo prove Normal convergence of the Weierstrass elliptic-function series, let $K\subset \mathbb{C}\setminus \Lambda$ be compact and choose $R$ with $|z|\le R$ on $K$. For $|\omega |>2R$,The comparison series ${\sum }_{\omega \ne 0}|\omega {|}^{-3}$ converges by the stated lattice-sum criterion. The Weierstrass M-test gives uniform convergence on $K$; since $K$ was arbitrary, the series converges locally uniformly away from $\Lambda$. The locally uniform convergence of holomorphic functions makes its sum holomorphic there, and the displayed principal term gives a double pole at every lattice point.

The function $\wp$ is even and satisfiesThe three nonzero two-torsion points $z_1,z_2,z_3$ are distinct modulo $\Lambda$ and each equals its own negative modulo $\Lambda$. Therefore no two can have the same $\wp$-value, so $e_1,e_2,e_3$ are distinct. Equivalently, $\wp$ is a degree-two branched map and each half-lattice point already occupies its fiber with multiplicity two.

For a transitive example, take the equianharmonic latticeand define $\theta ([z])=[\omega z]$. Multiplication by $\omega$ preserves $\Lambda$, so $\theta :\mathbb{C}/\Lambda \to \mathbb{C}/\Lambda$ is a conformal equivalence. On the nonzero half-lattice points it acts bybecause ${\omega }^2=-1-\omega$. Thus it acts transitively.

For each nonzero period lattice point $\omega$, the binomial series givesPairing $\omega$ with $-\omega$ cancels every odd power. Hence the Laurent series of the Weierstrass elliptic function iswhich is the stated coefficient formula.

Lift $u$ through the complex exponential by settingThe conformal invariance of harmonicity makes $v$ harmonic. Since $e^{w+2\pi i}=e^w$ and $u(2z)=u(z)$,Thus $v$ is bounded on the compact rectangleand its two periods make it bounded on the whole plane. The harmonic Liouville theorem makes $v$ constant. Since the exponential map is surjective onto ${\mathbb{C}}^{\ast }$, $u$ is constant.

For the requested counterexample after deleting a countable set, takeand let ${\wp }_{\Lambda }$ be the Weierstrass elliptic function for the latticeDefineChanging a branch of the logarithm adds $2\pi i$, a period of ${\wp }_{\Lambda }$, so $u$ is well defined. Its poles project precisely to $S$, and the conformal invariance of harmonicity makes $u$ harmonic away from them. The other period givesMoreover, $z\notin S$ implies $2z\notin S$. Finally, $u$ is nonconstant because as positive real $z\to 1$,which is real and unbounded. This is a scale-periodic harmonic function from an elliptic function.

If $F:X\to Y$ is a nonconstant holomorphic map between compact connected Riemann surfaces, its local degree at $p$ is the integer $m_F(p)\ge 1$ for which suitable local coordinates giveThe valency theorem states thatis independent of $y\in Y$. This common value is the degree of a holomorphic map, denoted $\mathrm{deg}F$.

Now let $f$ be a nonconstant rational function of degree $d$. If its distinct finite poles have orders $m_1,\dots ,m_r$, and its pole order at infinity is $m_{\infty }\ge 0$, thenThe derivative $f^{\prime }$ has a pole of order $m_j+1$ at each finite pole. When $m_{\infty }>0$, the expansion $f(z)\sim c{z}^{m_{\infty }}$ shows that $f^{\prime }(z)\sim c{m}_{\infty }{z}^{m_{\infty }-1}$ at infinity. Hence the degree of the derivative of a rational function isIn the first case $r\ge 0$, while in the second $1\le r\le d$; thereforeFor every $d\ge 1$, the lower bound is attained by $f(z)=z^d$, whose derivative has degree $d-1$ (with a constant assigned degree zero). For distinct $a_1,\dots ,a_d$, the functionhas $d$ simple poles, degree $d$, and a derivative with $d$ double poles, so $\mathrm{deg}{f}^{\prime }=2d$. Thus both rational bounds are sharp for every $d$.

Let next $g$ be a nonconstant elliptic function for the period lattice $\Lambda$. Its degree is the total order of its poles in a fundamental parallelogram; by the valency theorem, this is also the degree of the induced map $\mathbb{C}/\Lambda \to \hat{\mathbb{C}}$. If these poles have orders $n_1,\dots ,n_s$, then $d=\mathrm{deg}g={\sum }_j{n}_j$, and $g^{\prime }$ has poles of orders $n_j+1$. The degree of the derivative of an elliptic function is consequentlySince every nonconstant elliptic function has at least one pole and $1\le s\le d$,

Let $d\ge 3$ be odd. The Weierstrass elliptic function supplies the lower-bound exampleIt has one pole modulo $\Lambda$, of order $d$, so its derivative has one pole of order $d+1$. For the upper bound, choose distinct points $a_1,\dots ,a_d$ modulo $\Lambda$ and nonzero constants $c_j$ with ${\sum }_j{c}_j=0$. The quasi-periodicity of the Weierstrass zeta function makeselliptic. It has exactly $d$ simple poles, while $g^{\prime }$ has $d$ double poles. Therefore the two bounds are attained for every required odd degree.

For real $0<k<1$, the period lattice of $\mathrm{sn}(u,k)$ is generated byThe periods arise by composing the monodromy reflections of its inverse elliptic integral of the first kind.

Let ${\wp }_{\Lambda }$ be the Weierstrass elliptic function forBecause $2\pi i$ is a period, the formulais independent of the branch of the complex logarithm. It is a nonconstant harmonic function onand its other period gives $u(qz)=u(z)$.

For an integral alternating form $E$ arising as the imaginary part of a Hermitian form, a semicharacter satisfies $\alpha (\lambda +\mu )=e^{\pi iE(\lambda ,\mu )}\alpha (\lambda )\alpha (\mu )$. The Appell–Humbert theorem uses this datum together with the Hermitian form to specify a holomorphic line bundle on a complex torus. The ratio of two semicharacters is an ordinary unitary group homomorphism. On the square period lattice, with $H(z,w)=z\overline{w}$, all semicharacters are $\alpha (m+ni)=(-1{)}^{mn}{u}^m{v}^n$ for $u,v\in U(1)$.

The Weierstrass elliptic function satisfiesWriting $G_{2r}={\sum }_{\omega \in \Lambda \setminus \{0\}}{\omega }^{-2r}$, the Laurent coefficients of the Weierstrass elliptic function giveThus $g_2=60G_4$ and $g_3=140G_6$ cancel every nonremovable term in the Laurent series of ${\wp }^{\prime 2}-4{\wp }^3+g_2\wp +g_3$ at each point of the period lattice. The resulting elliptic function is an entire function and is bounded, hence is zero by Liouville theorem.

For a period lattice $\Lambda \subset \mathbb{C}$, the Weierstrass elliptic function is the normally convergent lattice sumIt is an even elliptic function with a double pole of principal part $(z-\omega {)}^{-2}$ and zero residue at every point $\omega$ of the period lattice.

For a nonzero elliptic function with period lattice $\Lambda$, its zeros of a function $a_j$ and poles $b_j$, repeated with their multiplicities, have equal total number and satisfy ${\sum }_j{a}_j-{\sum }_j{b}_j\in \Lambda$. Choose a fundamental parallelogram avoiding all zeros and poles on its boundary. The argument principle applied to $f^{\prime }/f$ gives equality of the numbers. Applying the residue theorem to $z{f}^{\prime }/f$ gives their difference of sums. Pairing opposite edges, the extra factors are the two lattice generators multiplied by integrals of $f^{\prime }/f$ along an edge. These integrals are integer multiples of $2\pi i$, because the endpoints have the same nonzero function value. Thus the difference of sums is a lattice element. This links zeros of a pulled-back line to the chord-and-tangent group law.