Elliptic function

An elliptic function is a meromorphic function with two real-linearly independent periods ${\omega }_1,{\omega }_2$. A fundamental cell is a half-open parallelogramOpposite boundary integrals cancel. Consequently a nonconstant elliptic function has equally many zeros and poles in a cell, counted with multiplicity, and the sum of its pole residues in a cell is zero.

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.

For a nonzero elliptic function, the weighted sum of its zero positions minus its pole positions belongs to the period lattice. Integrating $z{f}^{\prime }/f$ around a fundamental parallelogram proves this: translating opposite edges gives a lattice-linear combination of integrals of $f^{\prime }/f$, each an integer multiple of $2\pi i$. The unweighted integral proves that the zero and pole multiplicities have equal total.

Two real-linearly independent periods ${\omega }_1,{\omega }_2$ generate the period lattice

A rank-two period lattice $\Lambda \subset \mathbb{C}$ stable under multiplication by $i$ is, up to nonzero complex scaling, $\mathbb{Z}+i\mathbb{Z}$. Choose a shortest nonzero period $\omega$. Every complex number lies within $1/\sqrt{2}<1$ of a Gaussian integer, so for any $\eta \in \Lambda$, subtraction of a suitable $(m+ni)\omega$ leaves a shorter period and hence zero. This proves $\Lambda =\omega \mathbb{Z}[i]$.

Similarly, stability under a primitive cube root $\zeta$ gives $\Lambda =\omega (\mathbb{Z}+\zeta \mathbb{Z})$. The triangular lattice has covering radius $1/\sqrt{3}<1$, so the same shortest-period argument applies. Stability under a primitive sixth root is equivalent, since $e^{\pi i/3}=1+e^{2\pi i/3}$. These facts identify complex tori with automorphisms fixing their origin of orders four and three or six, respectively.

A fundamental parallelogram for $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$ is a translate ofIts translates tile the complex plane.

If an elliptic function had no poles, it would be entire. It is bounded on the closure of a fundamental parallelogram, and periodicity then makes it bounded on the whole complex plane. The Liouville theorem would make it constant. Thus every nonconstant elliptic function has a pole.

The Jacobi elliptic functions include $\mathrm{sn}(u,k)$, $\mathrm{cn}(u,k)$ and $\mathrm{dn}(u,k)$, with ${\mathrm{sn}}^2+{\mathrm{cn}}^2=1$ and ${\mathrm{dn}}^2+k^2{\mathrm{sn}}^2=1$. Their ratios give the other standard members. The Jacobi elliptic sine is the inverse of the elliptic integral of the first kind.

The Jacobi elliptic sine is the local inverse of the elliptic integral of the first kind: ifthen $z=\mathrm{sn}(u,k)$. Its analytic continuation is a meromorphic doubly periodic function.

It is the sn member of the Jacobi elliptic functions.

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.

For any finite value $a$, apply the argument principle to $f-a$ around a fundamental parallelogram whose boundary avoids zeros and poles. Periodicity cancels the opposite-edge integrals, so the number of solutions of $f(z)=a$ equals the fixed number of poles of $f$, counting multiplicities.

Associated with a period lattice are the Weierstrass sigma, zeta, eta and elliptic functions. They satisfy $\zeta ={\sigma }^{\prime }/\sigma$ and $\wp =-{\zeta }^{\prime }$; the eta constants record quasi-period increments of the Weierstrass zeta function.

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.

On its period torus, the derivative has one triple pole, so degree three. The Riemann-Hurwitz formula gives total ramification six, counting each ramification point with weight $e_p-1$. The number of distinct ramification points is five when $g_2\ne 0$ and three when $g_2=0$. Indeed ${\wp }^{\prime \prime }=6{\wp }^2-g_2/2$ by the Weierstrass elliptic differential equation. If $g_2\ne 0$, each of its two possible $\wp$-values has two distinct unramified preimages for $\wp$, because coincidence with a cubic branch root would force a repeated root. These four simple zeros of ${\wp }^{\prime \prime }$ give local degree two for ${\wp }^{\prime }$, and its triple pole contributes local degree three. If $g_2=0$, nonsingularity forces $g_3\ne 0$; the two simple zeros of $\wp$ are double zeros of ${\wp }^{\prime \prime }$ and each gives local degree three for ${\wp }^{\prime }$. Along with its pole these are three distinct points, each of weight two. A hexagonal period lattice realizes this second case by its sixth-root rotational symmetry.

For the square period lattice, multiplying the defining series by the lattice symmetry $z\mapsto iz$ gives $\wp (iz)=-\wp (z)$. At $h=(1+i)/2$, one has $ih=h-1$, so periodicity gives $\wp (h)=-\wp (h)$ and hence the displayed zero. This is one of the three finite branch values of the degree-two Weierstrass elliptic function on its torus. The three values are always distinct but need not all be nonzero. Distinctness is enough for the nonsingularity of the associated elliptic curve $Y^{2}Z=4{\prod }_i(X-e_{i}Z)$.

The Weierstrass elliptic function satisfies the displayed meromorphic function identity. To prove it, the Riemann-Hurwitz formula shows that its derivative has exactly three simple zeros modulo its period lattice, at the nonzero two-torsion points of a complex torus. At each such half-period $h$, evenness around $h$ gives ${\wp }^{\prime \prime }/{\wp }^{\prime }=1/(z-h)+O(z-h)$, so the right side has principal part $1/(4(z-h{)}^2)$, matching $\wp (2z)$. At zero, $\wp =z^{-2}+O(z^2)$ and ${\wp }^{\prime \prime }/{\wp }^{\prime }=-3/z+O(z^3)$ give the same principal part and zero constant term. The difference is a holomorphic elliptic function, hence constant by compactness, and its limit at zero is zero.

For generic fixed $w$, regard the right side as an elliptic function of $z$. The apparent pole at zero cancels, giving limiting value $\wp (w)$. At $z=w$ the slope ratio is finite. At $z=-w$, its squared slope has the same double principal part as $\wp (z+w)$ and no simple pole term. These exhaust the possible poles because the Weierstrass elliptic function has degree two. The difference is therefore an entire elliptic function, hence constant by Liouville theorem, and its value at zero is zero. Meromorphic continuation extends the identity to the exceptional parameters.

For a period lattice, the displayed canonical product converges normally because its logarithmic tails are bounded by a constant times $|\omega {|}^{-3}$. It defines an odd entire function with simple zeros exactly at the lattice and derivative one at zero. Its logarithmic derivative is the Weierstrass zeta function, and the derivative of that logarithmic derivative is minus the Weierstrass elliptic function. Sigma is quasi-periodic rather than elliptic; balanced products of its translates construct meromorphic functions with prescribed divisors on a complex torus.

For an oriented lattice basis with $\mathrm{Im}({\omega }_2/{\omega }_1)>0$, write ${\eta }_j=\zeta (z+{\omega }_j)-\zeta (z)$. These differences are constant because ${\zeta }^{\prime }=-\wp$ is periodic. Integrating zeta around a fundamental cell gives the displayed relation by its one simple pole of residue one. Oddness and the logarithmic derivative then give $\sigma (z+{\omega }_j)=-e^{{\eta }_j(z+{\omega }_j/2)}\sigma (z)$. This fixes both the sign and the exponent convention in sigma-product constructions.

The Weierstrass elliptic function has degree two on its period torus. Its three distinct half-period values $e_j$ have double local preimages, and its pole is double, so these four values are totally ramified. This also follows from ${\wp }^{\prime 2}=4{\prod }_j(\wp -e_j)$ and the principal part at the pole. A Möbius transformation with pole away from these values moves all four to finite values without changing local degrees.

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.

For $2a\notin \Lambda$, the apparent double poles at $z=\pm a$ on the left are canceled by its squared zero factor. Subtract the right side: the order-three principal parts at lattice points cancel by the even Laurent expansion of $\wp$, leaving at most a simple pole per period cell. An elliptic function has total residue zero in a period cell, so that possible pole is removable. The resulting entire periodic function is constant, and its oddness makes the constant zero. This proves the identity without presupposing the elliptic differential equation.

The determinant of the columns $(1,\wp (z),{\wp }^{\prime }(z))$, $(1,\wp (w),{\wp }^{\prime }(w))$ and $(1,\wp (-z-w),{\wp }^{\prime }(-z-w))$ vanishes identically. Its apparent cubic poles at $z=0,-w$ have order at most two because the coefficient multiplying the cubic-pole entry vanishes linearly. Column coincidences give six zeros, unless $3w$ is a lattice point, when they give four zeros with one multiple zero. In the half-period case, the two apparent zero locations at poles are removable zeros: ${\wp }^{\prime }(w)=0$ and the two $1/z$ terms cancel, leaving $O(z)$. Thus the zero count exceeds possible pole order in every case. The argument principle proves the identity, which expresses collinearity of the three points on the Weierstrass equation of an elliptic curve.

For distinct finite point classes on a complex elliptic curve, ordinary column determinants test whether a line meets the curve at those three points. Repeated columns instead vanish for a purely linear-algebraic reason. To prescribe a tangent intersection at $u$, replace the second equal column by its derivative; to prescribe a triple intersection, use $c(u),c^{\prime }(u),c^{\prime \prime }(u)$. These confluent determinants vanish exactly when the corresponding degree-three intersection divisor has point sum zero, away from the affine chart's pole. A local homogeneous lift supplies the version at the origin.

Every elliptic function has a unique expression $A(\wp )+B(\wp ){\wp }^{\prime }$ with rational functions $A,B$. Its even part is rational in the Weierstrass elliptic function; its odd part divided by ${\wp }^{\prime }$ is even and meromorphic, so is also rational in $\wp$. The local even Laurent series at a half-period and at the pole justify meromorphic descent through the double cover of the Riemann sphere.

The Weierstrass elliptic function is the quotient map from the complex torus by $z\mapsto -z$. An even elliptic function descends through this map; its even Laurent series at each branch point makes the descended function meromorphic there. A meromorphic function on the Riemann sphere is rational, so $f=Q(\wp )$.

An even elliptic function descends through the degree-two quotient map $\wp$ to a rational function $R$ on the Riemann sphere. If the original function also has degree two, composition of map degrees gives $2=2\mathrm{deg}R$, so $R$ has degree one. Consequently it is a Möbius transformation with $ad-bc\ne 0$. The shift $\mathrm{sn}(z+K,k)$ is even and has degree two, providing an example for the common period lattice of the Jacobi elliptic sine and $\wp$.

On every compact set disjoint from the lattice, the summandis bounded by $C|\omega {|}^{-3}$ for all sufficiently large $|\omega |$. Since the lattice sum of $|\omega {|}^{-3}$ converges, the defining series for $\wp$ converges normally.

At the three nonzero two-torsion points $z_i$ of $\mathbb{C}/\Lambda$, put $e_i=\wp (z_i)$. The values are distinct because $\wp (z)=\wp (w)$ exactly when $z\equiv \pm w(\mathrm{mod}\Lambda )$.

Let $h_i$ represent a nonzero two-torsion point of a complex torus, and write $e_i=\wp (h_i)$. The product $(\wp (z+h_i)-e_i)(\wp (z)-e_i)$ is an elliptic function whose apparent poles cancel against double zeros. It is therefore constant by Liouville theorem. Its value at $z=0$ is ${\wp }^{\prime \prime }(h_i)/2$. The Weierstrass elliptic differential equation gives ${\wp }^{\prime \prime }(h_i)/2=3e_i^2-g_2/4=(e_i-e_j)(e_i-e_l)$, proving the formula.

At $z=h_i/2$, evenness and periodicity give $\wp (z+h_i)=\wp (z)$. The Weierstrass half-period translation formula therefore identifies $(\wp (h_i/2)-e_i{)}^2$ with $(e_i-e_j)(e_i-e_l)$. Differentiating the Weierstrass half-period translation formula proves the identity, interpreted as equality of meromorphic functions, including the apparent singularities of its quotients.

A two-torsion point satisfies $2z\in \Lambda$. The three nonzero classes are represented by half-periods $\lambda /2$, $\mu /2$, and $(\lambda +\mu )/2$.

The equianharmonic triangular lattice is preserved by multiplication by the cube root of unity $\omega =e^{2\pi i/3}$. On its three nonzero two-torsion classes, multiplication by $\omega$ acts as a three-cycle.

The Weierstrass zeta function satisfies ${\zeta }^{\prime }=-\wp$ and is quasi-periodic: for every period $\omega$, the difference $\zeta (z+\omega )-\zeta (z)$ is constant. It has a simple pole of residue one at each lattice point. Consequently, if ${\sum }_j{c}_j=0$, thenis an elliptic function.

It is the quasi-periodic zeta member of the Weierstrass functions.

Writing $G_{2r}={\sum }_{\omega \ne 0}{\omega }^{-2r}$ for the lattice Eisenstein sums, expansion about zero givesThus the constant coefficient vanishes and, for $k\ge 1$, the coefficient of $z^{2k}$ is $(2k+1)G_{2k+2}$.

Write the Weierstrass elliptic function as $\wp (z)=z^{-2}+{\sum }_{n\ge 1}{c}_n{z}^{2n}$, with $c_1=3G_4$, $c_2=5G_6$ and $c_n=(2n+1)G_{2n+2}$. Differentiate the Weierstrass elliptic differential equation to obtain ${\wp }^{\prime \prime }=6{\wp }^2-g_2/2$. Comparing coefficients of $z^{2n-2}$ gives the displayed recurrence for $n\ge 3$. Every denominator is positive, so induction expresses every higher lattice Eisenstein series as a polynomial in $G_4,G_6$ with positive rational coefficients on its occurring monomials. In particular $G_8=3G_4^2/7$ and $G_{10}=5G_4{G}_6/11$.

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.

The three nonzero half-period classes make the odd derivative of the Weierstrass elliptic function vanish. Their values are distinct: two equal values would force at least four zeros for a function with only one double pole. Thus they are three distinct roots of $4X^3-g_{2}X-g_3$, whose monic discriminant gives the displayed factor. This lattice discriminant differs by $(2\pi {)}^{12}$ from the normalized modular discriminant for the lattice $\mathbb{Z}+\mathbb{Z}\tau$.