In this section $f$ is an order 2 elliptic function with respect to some fixed lattice $\Omega$. It is assumed to have two distinct simple poles placed symmetrically about the origin.

Even Symmetry

Let $f(z)$ be an elliptic function of order 2 with two simple poles at $\pm\rho$ with residues $\pm\nu$ and $\rho \ne 0$ and $\nu \ne 0$. Then $f(z)$ and $f(-z)$ have Laurent power series expansions at $z = \rho$ with the same initial term

from which it follows that the elliptic function $f(z) - f(-z)$ has no pole at $z = \rho$. By a similar argument it has no pole at $z = -\rho$ and therefore it has no poles at all and is a constant. By evaluating at $z = 0$ it follows that this constant is zero and that

which is to say that $f$ is an even function if the midpoint of it's two poles lies at the origin.

Differential Equation

By considering the power series expansions of $f^n(z)$ at $z = \rho$ it can be seen that the set of functions $\{1, f, f^2, f^3, f^4\}$ is a basis for the vector space $\mathcal{V}$ of all even elliptic functions with poles of order at most 4 at $\{\rho,-\rho\}$ and no poles elsewhere.

By differentiating power series power it follows that $f'$ has poles at $\pm\rho$ of order 2. And by differentiating identity even it follows that $f'$ is an odd function.

Therefore $f'(z)^2 \in \mathcal{V}$ and can be written as a linear combination of its basis elements

This is an ordinary first order differential equation for $f$. From the power series expansion at $z = \rho$ it is easily determined that $\nu = \pm1 / \sqrt{a}$.

Sigma Product For $f$

By considering $f(z_1)-f(z_2)$ as an order 2 elliptic function in $z_1$ with poles at $\pm\rho$ and zeroes at $\pm z_2$ we can determine that the ratio of the two sides of the equation below has no poles or zeroes and is therefore a constant.

Further that constant can be obtained by considering the power series expansion of both sides at $z_1 = \rho$.

Sigma Product For $f'$

Dividing sigma_product_f by $z_1 - z_2$ and then taking the limit as $z_2 \rightarrow z_1$ yields this sigma product for $f'(z)$

Using sigma_product_f and sigma_product_df and the duplication formula for $\sigma$

where $\omega_1,\omega_2$ are distinct half-periods and $\omega_1 + \omega_2 + \omega_3 = 0$ and $\omega_4 = 0$, it is possible to verify a second form of the differential equation for $f$

From this it follows that if $e_i$ are the four roots of the polynomial $ax^4 + bx^3 + cx^2 + dx + e = 0$ then $e_i = f(\omega_i)$.

Cross-Ratio Identity Formula

Substituting formula sigma_product_f into the cross ratio formula, all the poles and constants cancel to leave

This coinicides nicely with the fact that both the cross-ratio and differential equation are invariant under Möbius transformations.

Monomial Basis Sequence

Similar to the basis $\{1,f,f^2,f^3,f^4\}$ for even functions constructed above, $\{1,f,f^2,f'\}$ is a basis for the vector space of all (not necessarily even) elliptic functions with poles of order at most 2 at ${\rho,-\rho}$. We can extend this basis indefinitely by adding two elements $f^n$ and $f^{n-2}f'$ for $n = 2,3 \ldots$ to the initial basis $\{1,f\}$.

The fact that all elements are monomials is not crucial to the following arguments but does have the advantage of generating considerably simpler formulae.

Symmetric Addition Formula

Using the first 4 elements of monomial_basis we can apply the extended Frobenius-Stickelberger formulae to obtain

From symmetric_sigma we obtain the symmetric addition formula for $f$

Polynomial Resultant Identity

The following identity is an elliptic function analog of an algebraic identity which arises when solving the quartic curve / parabola intersection equation. It can be directly verified by expanding the LHS as a sigma product using sigma_product_f and the RHS as a sigma product using symmetric_sigma.

where $K$ is a function of $z_1,z_2,z_3$ but not of $z$. Letting $z \rightarrow \rho$ in poly_res_f gives

Taking the product of poly_res_f over $z \in \{0, \omega_1,\omega_2,\omega_3\}$ and using de_2 gives

Addition Formulae In 3 Variables

Putting $z$ equal to a zero of $f$ in equation poly_res_f and using poly_res_f_K yields the following three variable addition formula for $f(z)$

Similarly taking the square root of poly_res_df, with the correct sign being determined by guess work, gives a three variable addition formula for $f'(z)$

where $e_1,e_2,e_3,e_4$ are the roots of the polynomial equation $ax^4 + bx^3 + cx^2 + dx + e = 0$.

These formulae express $f(z_1+z_2+z_3)$ and $f'(z_1+z_2+z_3)$ as a rational functions of $f(z_1),f'(z_1),$ $f(z_1),f'(z_2),$ $f(z_3),f'(z_3)$ and $a,b,c,d,e$. The latter five being rational because the algebraic expressions involving them are symmetric.

Double Pole Case

Putting $a = 0$ in addition_f gives

Putting $e_4 = -b/a$ in addition_df, performing some row operations, letting $a \rightarrow 0$, and then cancelling out a common determinant, gives

where $e_1,e_2,e_3$ are the roots of the polynomial equation $bx^3 + cx^2 + dx + e = 0$.

In this section $g$ is an order 2 elliptic function with respect to some fixed lattice $\Omega$. It is assumed to have two distinct simple poles in an arbitrary location.

Arbitrary Order 2 Elliptic Functions

All the previous formulae remain true in some form for arbitrary order 2 elliptic functions. Let $g$ be such a function and $\rho_1,\rho_2$ be its poles with residues $\pm\nu$. Then $g$ may be written very simply in terms of an even function $f$ by translating it's argument to the midpoint $T$ of the two poles as follows

where $T = -\tfrac 1 2 (\rho_1 + \rho_2)$ and $\rho = \tfrac 1 2 (\rho_1 - \rho_2)$. Note that $g(z)$ also satisfies differential equation de.

Now it is just a matter of going through the previous equations and replacing $\rho$ by $\tfrac 1 2 (\rho_1 - \rho_2)$ and $z_i$ by $z_i + T$ followed by replacing $f(z_i + T)$ by $g(z_i)$.

The evenness property even becomes

The sigma product sigma_product_f becomes

The symmetric addition formula symmetric_sigma becomes

Addition Formula In 2 Variables

Finally the three variable addition formula for even order 2 elliptic functions can be converted into a two variable addition formulae for arbitrary order 2 elliptic functions.

For equation addition_f the process outlined above does not quite work out because on the LHS we are left with $f(z_1 + z_2 + z_3 + 3T)$ instead of the desired $f(z_1 + z_2 + z_3 + T)$. This can be remedied by changing the mapping to

and noting using f_g and even that $f(z_3) = f(-T) = f(T) = g(0)$ and $f'(z_3) = f'(-T) = -f'(T) = -g'(0)$ introducing a minus sign on all the $f'(z_3)$ term's and yielding

and for the derivative we have

These formulae express $g(u+v)$ and $g'(u+v)$ as a rational functions of $g(u),g'(u),$ $g(v),g'(v),$ $g(0),g'(0)$ and $a,b,c,d,e$.

Limit Cases

Equations addition_g and addition_dg need clarification when $g$ or $g'$ has a zero or a pole at $z=0$. If $g(0) = 0$ then

References

M. Popoviciu Draisma 2014 Invariants of binary forms. http://dx.doi.org/10.5451/unibas-006268669 Doctoral Dissertation: Philosophisch-Naturwissenschaftlichen Fakultät der Universität Basel