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For the period lattice $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$, define the Weierstrass elliptic function byOn a compact set away from $\Lambda$, the summand is $O(|\omega {|}^{-3})$ uniformly for sufficiently large $|\omega |$. The permitted lattice convergence estimate therefore gives Normal convergence of the Weierstrass elliptic-function series. Consequently $\wp$ is a meromorphic function, with a double pole of principal part $(z-\omega {)}^{-2}$ at each lattice point and no other poles. Termwise differentiation gives the normally and absolutely convergent seriesPairing $\omega$ with $-\omega$ shows that $\wp$ is even and ${\wp }^{\prime }$ is odd. Reindexing the absolutely convergent derivative series gives ${\wp }^{\prime }(z+\lambda )={\wp }^{\prime }(z)$ for every $\lambda \in \Lambda$. Thus $\wp (z+\lambda )-\wp (z)$ is constant, with removable singularities at the lattice points. For either basis period $\lambda ={\omega }_1,{\omega }_2$, evaluate at $z=-\lambda /2$: evenness makes this constant zero. These two periods imply all periods in $\Lambda$. Hence $\wp$ is a nonconstant elliptic function with period lattice $\Lambda$.

To obtain the Laurent coefficients of the Weierstrass elliptic function, take $0<|z|<{\min }_{\omega \ne 0}|\omega |$ and expandThe expansion and lattice summation can be interchanged uniformly on smaller disks, using the same convergent majorant. The odd powers cancel in pairs. In terms of the lattice Eisenstein sums, the answer isOnly the absolutely convergent lattice Eisenstein sums of weights at least four occur here; no conditionally defined $G_2$ is needed.

Put $g_2=60G_4$ and $g_3=140G_6$. The Laurent series just found givesTherefore $F={\wp }^{\prime 2}-4{\wp }^3+g_2\wp +g_3$ has no pole at zero, and its value there is zero. By periodicity it has no poles anywhere. An entire elliptic function is bounded on a closed fundamental parallelogram, hence on the plane, and is constant by Liouville theorem. Since $F(0)=0$, this proves the Weierstrass elliptic differential equationWe still need to identify the three roots without assuming their distinctness. For a nonconstant elliptic function $u$, the argument principle applied to $u^{\prime }/u$ around a fundamental parallelogram makes the opposite-edge integrals cancel. Thus its zeros and poles, counted with multiplicity, have equal total number. In particular $\wp -a$ has exactly two zeros in a period cell for every finite $a$.

Let $h_i={\omega }_i/2$, with ${\omega }_3=-{\omega }_1-{\omega }_2$. Their classes are the three nonzero two-torsion points of a complex torus. Oddness and periodicity give ${\wp }^{\prime }(h_i)=-{\wp }^{\prime }(h_i)$, so ${\wp }^{\prime }(h_i)=0$. Thus $\wp -e_i$, where $e_i=\wp (h_i)$, has a zero of multiplicity at least two at $h_i$, and the preceding count says its multiplicity is exactly two. If two of the half-period values of the Weierstrass elliptic function were equal, the same function would have at least four zeros, a contradiction. Evaluating the Weierstrass elliptic differential equation at the three half-periods now gives three distinct roots of its cubic. Consequently

For the remaining identity, fix $i$ and put $A_i=(e_i-e_j)(e_i-e_l)$, where $\{i,j,l\}=\{1,2,3\}$. The elliptic functionhas possible poles only at $z=0,-h_i$ modulo $\Lambda$. At each, the other factor has an exact double zero, so these poles are removable. Thus $H_i$ is constant, and its limit at zero is ${\wp }^{\prime \prime }(h_i)/2$. Differentiating the cubic differential equation first where ${\wp }^{\prime }\ne 0$, then extending the resulting identity holomorphically, gives ${\wp }^{\prime \prime }=6{\wp }^2-g_2/2$. Since $e_i$ is a root of $P(X)=X^3-g_{2}X/4-g_3/4$, we obtainThis proves the Weierstrass half-period translation formula by pole cancellation. Set $u_i=h_i/2={\omega }_i/4$. Since $u_i+h_i\equiv -u_i(\mathrm{mod}\Lambda )$, evenness gives $\wp (u_i+h_i)=\wp (u_i)$, and thereforeThe denominator here is nonzero: the only zero of $\wp -e_i$ modulo $\Lambda$ is $h_i$, whereas $u_i$ is not congruent to $h_i$. Finally differentiate the Weierstrass half-period translation formula and use $z-h_i\equiv z+h_i(\mathrm{mod}\Lambda )$ to obtain the Weierstrass quarter-period derivative identityThe equality holds wherever these quotients are initially defined, and hence as an identity of meromorphic functions at all remaining points.

Write the period lattice as $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$, with ${\omega }_1,{\omega }_2$ real-linearly independent and $\mathrm{Im}({\omega }_2/{\omega }_1)>0$. A meromorphic function $f$ is an elliptic function for this lattice if $f(z+\lambda )=f(z)$ for every $\lambda \in \Lambda$. If it has no poles, it is an entire function bounded on the closure of a fundamental parallelogram. Periodicity makes it bounded on all of $\mathbb{C}$, so the Liouville theorem proves that an elliptic function without poles is constant.

Here are the two contour identities needed for counting its zeros and their sum. Choose a positively oriented fundamental parallelogram $P$ whose boundary avoids all zeros and poles, and put $h=f^{\prime }/f$. The argument principle givesbecause $h$ is periodic and opposite edges cancel. Here $N,P_f$ count zeros and poles with multiplicity. For the weighted integral, let $J_i$ be the integral of $h$ along the edge from a vertex $c$ to $c+{\omega }_i$. Translating the opposite edges givesThe endpoint values of $f$ agree along either edge. A continuous logarithm along the edge therefore changes by $2\pi i{n}_i$, with $n_i\in \mathbb{Z}$, and $J_i=2\pi i{n}_i$. On the other hand, the residue theorem evaluates the left side as the sum of the zero positions minus the sum of the pole positions, both with multiplicity. Thus the elliptic divisor-sum identity isUnder the stated pole hypothesis there is just one pole class, represented by zero, of order $m$. ConsequentlyChanging representatives of the zero classes changes their weighted sum by an element of $\Lambda$, so the congruence is well defined.

The Weierstrass elliptic function isOn a bounded set of $z$, the summand is $O(|\omega {|}^{-3})$ for large $|\omega |$. The lattice sum of $|\omega {|}^{-3}$ converges in real dimension two, proving Normal convergence of the Weierstrass elliptic-function series on compact sets away from $\Lambda$. Hence $\wp$ is a holomorphic function there, and at zero its principal part is $z^{-2}$, so it has a double pole. Replacing $\omega$ by $-\omega$ shows that $\wp$ is even. Differentiating the normally convergent series giveswhose absolute convergence permits reindexing by any $\lambda \in \Lambda$. Thus ${\wp }^{\prime }$ is periodic and $\wp (z+\lambda )-\wp (z)$ is constant. Substituting $-z-\lambda$ for $z$ and using evenness changes that constant to its negative, so it is zero. This proves that $\wp$ is an elliptic function, with double poles precisely at the lattice points.

The preceding zero count applied to $\wp -a$ says that it has exactly two zeros modulo $\Lambda$, counted with multiplicity. Evenness pairs any zero $z$ with $-z$. If these are distinct classes they must both be simple. If they coincide, $2z\in \Lambda$, and locally $\wp (z+t)=\wp (z-t)$: the zero has even order, which the total count forces to be two. In particular such a zero cannot be a lattice point, where $\wp$ has a pole. Every finite fibre of the Weierstrass function consists of two opposite simple points or one double nonzero half-period point.

To obtain the rational representation, first consider an even elliptic function $g$. The preceding description proves that $\wp :\mathbb{C}/\Lambda \to {\mathbb{P}}^1$ has exactly the fibres $\{z,-z\}$, so $g$ defines a function $A$ of $w=\wp (z)$. Away from the branch values a local inverse of $\wp$ makes $A$ meromorphic. At a half-period $z_0$, the local expansion is $w-w_0=c{t}^2+O(t^4)$, with $c\ne 0$ because the zero has order exactly two. The Laurent series of $g(z_0+t)$ contains only even powers, and $t^2$ is a holomorphic local coordinate as a function of $w-w_0$. Thus $A$ is meromorphic at that value too. At zero the same argument uses $1/w=t^2+O(t^6)$, proving meromorphy at infinity. A meromorphic function on the Riemann sphere is a rational function: subtract its finitely many principal parts and use compactness to make the remainder constant. This proves that even elliptic functions are rational in the Weierstrass function.

For an arbitrary elliptic function, split $f=f_{+}+f_{-}$, where $f_{\pm }(z)=(f(z)\pm f(-z))/2$. The even part is $A(\wp )$. Since ${\wp }^{\prime }$ is odd and not identically zero, $f_{-}/{\wp }^{\prime }$ is an even meromorphic elliptic function, including at the zeros of ${\wp }^{\prime }$ where the quotient may have poles. It is therefore $B(\wp )$ for a rational function $B$. The elliptic function-field decomposition isThe representation is unique: its even and odd parts determine $A$ and $B$, and $\wp$ takes every value on the Riemann sphere.

Let $\pi :\mathbb{C}\to \mathbb{C}/\Lambda$ be the universal covering map. Since the complex plane is a simply connected domain, the lifting criterion for a covering space gives a holomorphic lift between one-dimensional complex tori $F:\mathbb{C}\to \mathbb{C}$ such thatFor every $\lambda \in \Lambda$, the difference $F(z+\lambda )-F(z)$ belongs to the discrete period lattice $\Lambda$. It depends continuously on $z$, so it is constant. Differentiating shows that the derivative $F^{\prime }$ is $\Lambda$-periodic. It is bounded on the compact closure of a fundamental parallelogram, and periodicity makes it bounded on all of $\mathbb{C}$. The Liouville theorem therefore makes $F^{\prime }$ constant, and henceThus every holomorphic map has an affine lift of a holomorphic map between one-dimensional complex tori. If “map of complex tori” means an identity-preserving map, choose $F(0)=0$; then $b=0$, so $F$ is the required linear map. Without that convention the statement must say affine, since a nonzero translation in a group of the torus lifts to $z\mapsto z+b$.

The Weierstrass elliptic function of $\Lambda$ isThe subtracted term gives the Normal convergence of the Weierstrass elliptic-function series away from $\Lambda$. PutThe Laurent coefficients of the Weierstrass elliptic function at zero giveSet $g_2=60G_4$ and $g_3=140G_6$. Direct substitution shows that the principal part and constant term ofvanish at zero. Since $H$ is an elliptic function, translation gives the same cancellation at every point of the period lattice. Every apparent isolated singularity is therefore a removable singularity, so $H$ is an entire function. It is periodic and hence bounded on the translates of a compact fundamental parallelogram. The Liouville theorem gives $H=0$, which proves the Weierstrass elliptic differential equation

Finally suppose that $f$ is a biholomorphic group homomorphism. Its identity-preserving lift has the form $F(z)=\zeta z$ by the first part. Since the inverse map also lifts linearly,Choose a $\mathbb{Z}$-basis of $\Lambda$. Multiplication by $\zeta$ is then represented by a unimodular matrix $A\in {\mathrm{GL}}_2(\mathbb{Z})$. As a real-linear transformation of $\mathbb{C}$, it has determinant $|\zeta {|}^2>0$; hence $\det A=1$ and $|\zeta |=1$. The characteristic polynomial of $A$ and the Cayley-Hamilton theorem giveThis already has the required form with $m=-\mathrm{tr}(A)\in \mathbb{Z}$ and $n=1$. Moreover $2\mathrm{Re}\zeta =\mathrm{tr}(A)$, so $|\zeta |=1$ forces $\mathrm{tr}(A)\in \{-2,-1,0,1,2\}$. Thus $\zeta$ is a root of unity, of order $1$, $2$, $3$, $4$, or $6$, completing the description of an automorphism of a one-dimensional complex torus.

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.

The Normal convergence of the Weierstrass elliptic-function series allows its derivative to be written as an absolutely locally uniformly convergent sum of inverse cubes. For a period lattice $\Lambda$, this derivative sum is invariant under translation by any lattice vector, by reindexing. Consequently $\wp (z+\omega )-\wp (z)$ is constant; its apparent poles cancel. The Weierstrass elliptic function is even, so for a primitive lattice generator $\omega$, evaluation at $z=-\omega /2$ makes that constant zero. Applying this to two generators proves that this is a periodic function, namely $\wp$. This argument avoids separating the original convergent series into two divergent inverse-square sums.