Elementary Approach to Modular Equations: Jacobi's Transformation Theory 5

Jacobi's Second Real Transformation

The second transformation is obtained by taking $m = 0, m' = 1$ so that $\omega = iK'/p$. On the face of it the transformation thus involves imaginary quantities, but will be shown later to be a real transformation only. In the case of this transformation we will use $l_{1}$ in place of $l$ and $M_{1}$ in place of $M$. Also we will keep the factor $(-1)^{(p - 1)/2}$ with the multiplier $M_{1}$. We thus obtain the following by putting $\omega = iK'/p$ in the general formulas in $2s\omega$: $$\begin{align}\text{sn}\left(\frac{u}{M_{1}}, l_{1}\right) &= \frac{\text{sn}\,u}{M_{1}}\prod_{s = 1}^{(p - 1)/2}\dfrac{1 - \dfrac{\text{sn}^{2}\,u}{\text{sn}^{2}\,\dfrac{2siK'}{p}}}{1 - k^{2}\,\text{sn}^{2}\,u\,\text{sn}^{2}\,\dfrac{2siK'}{p}}\notag\\ &= \sqrt{\frac{k^{p}}{l_{1}}}\prod_{s = -(p - 1)/2}^{(p - 1)/2}\text{sn}\left(u + \frac{2siK'}{p}\right)\notag\\ \text{cn}\left(\frac{u}{M_{1}}, l_{1}\right) &= \text{cn}\,u\prod_{s = 1}^{(p - 1)/2}\dfrac{1 - \dfrac{\text{sn}^{2}\,u}{\text{sn}^{2}\left(K - \dfrac{2siK'}{p}\right)}}{1 - k^{2}\,\text{sn}^{2}\,u\,\text{sn}^{2}\,\dfrac{2siK'}{p}}\notag\\ &= \sqrt{\frac{l'_{1}k^{p}}{l_{1}k'^{p}}}\prod_{s = -(p - 1)/2}^{(p - 1)/2}\text{cn}\left(u + \frac{2siK'}{p}\right)\notag\\ \text{dn}\left(\frac{u}{M_{1}}, l_{1}\right) &= \text{dn}\,u\prod_{s = 1}^{(p - 1)/2}\dfrac{1 - k^{2}\,\text{sn}^{2}\,u\,\text{sn}^{2}\left(K - \dfrac{2siK'}{p}\right)}{1 - k^{2}\,\text{sn}^{2}\,u\,\text{sn}^{2}\,\dfrac{2siK'}{p}}\notag\\ &= \sqrt{\frac{l'_{1}}{k'^{p}}}\prod_{s = -(p - 1)/2}^{(p - 1)/2}\text{dn}\left(u + \frac{2siK'}{p}\right)\notag\\ M_{1} &= (-1)^{(p - 1)/2}\prod_{s = 1}^{(p - 1)/2}\left(\dfrac{\text{sn}\left(K - \dfrac{2siK'}{p}\right)}{\text{sn}\,\dfrac{2siK'}{p}}\right)^{2}\notag\\ l_{1} &= k^{p}\prod_{s = 1}^{(p - 1)/2}\text{sn}^{4}\left(K - \frac{2siK'}{p}\right)\notag\\ l'_{1} &= \dfrac{k'^{p}}{{\displaystyle\prod_{s = 1}^{(p - 1)/2}\text{dn}^{4}\,\frac{2siK'}{p}}}\notag\end{align}$$
Clearly the above formulas can be simplified by using the Jacobi' imaginary transformations to get rid of the imaginary unit $i$ and thus we obtain: $$\begin{align}\displaystyle l_{1} &= \dfrac{k^{p}}{{\displaystyle\prod_{s = 1}^{(p - 1)/2}\text{dn}^{4}\left(\dfrac{2sK'}{p}, k'\right)}}\notag\\ l'_{1} &= k'^{p}\prod_{s = 1}^{(p - 1)/2}\text{sn}^{4}\left(K - \frac{2sK'}{p}, k'\right)\notag\\ M_{1} &= \prod_{s = 1}^{(p - 1)/2}\left(\dfrac{\text{sn}\left(K - \dfrac{2sK'}{p}, k'\right)}{\text{sn}\left(\dfrac{2sK'}{p}, k'\right)}\right)^{2}\notag\\ \text{sn}\left(\frac{u}{M_{1}}, l_{1}\right) &= \frac{\text{sn}\,u}{M_{1}}\prod_{s = 1}^{(p - 1)/2}\dfrac{1 + \dfrac{\text{sn}^{2}\,u}{\text{sc}^{2}\left(\dfrac{2sK'}{p}, k'\right)}}{1 + k^{2}\,\text{sn}^{2}\,u\,\text{sc}^{2}\left(\dfrac{2sK'}{p}, k'\right)}\notag\end{align}$$ and similar formulas for $\text{cn},\,\text{dn}$.

From the last equation we can see that the least positive value of $u$ which the right side vanishes is $u = 2K$ and the left side vanishes for $u/M_{1} = 2L_{1}$ and finally we get the relation $K/M_{1} = L_{1}$.

Jacobi's Second Complementary Transformation

From the first two equations above we can see that $$\text{sc}\left(\frac{u}{M_{1}}, l_{1}\right) = \sqrt{\frac{k'^{p}}{l'_{1}}}\prod_{s = -(p - 1)/2}^{(p - 1)/2}\text{sc}\left(u + \frac{2siK'}{p}\right)$$ Replacing $u$ by $iu$ we get $$\begin{align}\text{sn}\left(\frac{u}{M_{1}}, l'_{1}\right) &= (-1)^{(p - 1)/2}\sqrt{\frac{k'^{p}}{l'_{1}}}\prod_{s = -(p - 1)/2}^{(p - 1)/2}\text{sn}\left(u + \frac{2sK'}{p}, k'\right)\notag\\ &= (-1)^{(p - 1)/2}\sqrt{\frac{k'^{p}}{l'_{1}}}\,\text{sn}(u, k')\prod_{s = 1}^{(p - 1)/2}\text{sn}\left(u + \frac{2sK'}{p}, k'\right)\text{sn}\left(u - \frac{2sK'}{p}, k'\right)\notag\\ &= \sqrt{\frac{k'^{p}}{l'_{1}}}\left(\prod_{s = 1}^{(p - 1)/2}\text{sn}\left(\frac{2sK'}{p}, k'\right)^{2}\right)\text{sn}(u, k')\notag\\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\prod_{s = 1}^{(p - 1)/2}\dfrac{1 - \dfrac{\text{sn}^{2}(u, k')}{\text{sn}^{2}\left(\dfrac{2sK'}{p}, k'\right)}}{1 - k'^{2}\,\text{sn}^{2}(u, k')\,\text{sn}^{2}\left(\dfrac{2sK'}{p}, k'\right)}\notag\end{align}$$ We can see that the factors on the right side of the above equation which are independent of $u$ must lead to the value $1/M_{1}$ (as can be verified by taking limits as $u \to 0$). Therefore we have $$\text{sn}\left(\frac{u}{M_{1}}, l'_{1}\right) = \frac{\text{sn}(u, k')}{M_{1}}\prod_{s = 1}^{(p - 1)/2}\dfrac{1 - \dfrac{\text{sn}^{2}(u, k')}{\text{sn}^{2}\left(\dfrac{2sK'}{p}, k'\right)}}{1 - k'^{2}\,\text{sn}^{2}(u, k')\,\text{sn}^{2}\left(\dfrac{2sK'}{p}, k'\right)}$$ From the above we can see that the least positive value of $u$ for which the right side vanishes is $u = 2K'/p$ and the left side vanishes for $u/M_{1} = 2L'_{1}$ so that we get $L'_{1} = K'/(pM_{1})$. From the equations $L_{1} = K/M_{1}$ and $L'_{1} = K'/(pM_{1})$ we get $K'/K = pL'_{1}/L_{1}$.

Finally from the Jacobi's first and second real transformations (and their complementary forms) we see that corresponding to a $k \in (0, 1)$ there are two unique moduli $l, l_{1} \in (0, 1)$ such that $l < k < l_{1}$ such that $$\frac{1}{p}\frac{L'}{L} = \frac{K'}{K} = p\frac{L'_{1}}{L_{1}}$$ Moreover the relation between $k, l$ is algebraic and relation between $k, l_{1}$ is also algebraic. If we analyze the forms of the Jacobi's transformations, we will see that the first complementary tranformation is analogous to the second transformation and second complementary transformation is analogous to the first transformation. From these considerations it follows that the the relation between $l, k$ is same as that between $k, l_{1}$ and the relation among $l, k, l_{1}$ is same as that among $l'_{1}, k', l'$.

Combining both the first and second transformations we can obtain the multiplication formulas for elliptic functions. For example, since the relation between $k, l_{1}$ is same as that between $l, k$ we can replace $k, l_{1}$ by $l, k$ in the second transformation and let $N_{1}$ denote the corresponding multiplier. Then we have $N_{1} = L/K = 1/(pM)$ so that $MN_{1} = 1/p$. The formula is now given by $$\text{sn}\left(\frac{u}{N_{1}}, k\right) = \frac{\text{sn}(u, l)}{N_{1}}\prod_{s = 1}^{(p - 1)/2}\dfrac{1 + \dfrac{\text{sn}^{2}(u, l)}{\text{sc}^{2}\left(\dfrac{2sL'}{p}, l'\right)}}{1 + l^{2}\,\text{sn}^{2}(u, l)\,\text{sc}^{2}\left(\dfrac{2sL'}{p}, l'\right)}$$ Replacing $u$ by $u/M$ we get $$\text{sn}(pu, k) = pM\,\text{sn}\left(\frac{u}{M}, l\right)\prod_{s = 1}^{(p - 1)/2}\dfrac{1 + \dfrac{\text{sn}^{2}\left(\dfrac{u}{M}, l\right)}{\text{sc}^{2}\left(\dfrac{2sL'}{p}, l'\right)}}{1 + l^{2}\,\text{sn}^{2}\left(\dfrac{u}{M}, l\right)\text{sc}^{2}\left(\dfrac{2sL'}{p}, l'\right)}$$ Using the first Jacobi transformation we can replace the functions of modulus $l$ on the right side with functions of modulus $k$ and thereby we obtain the multiplication formula for the elliptic function $\text{sn}$. We can obtain the multiplication formula in another way by using the first transformation to switch from modulus $k$ to a larger modulus $l_{1}$ and then using the second transformation to switch from modulus $l_{1}$ to the smaller modulus $k$.

To summarize Jacobi's transformations:
Given a modulus $k \in (0, 1)$ and a positive prime $p$ we have two unique moduli $l, l_{1} \in (0, 1)$ such that $l < k < l_{1}$ and
$$\frac{1}{p}\frac{L'}{L} = \frac{K'}{K} = p\frac{L'_{1}}{L_{1}}$$ Such a series of moduli $l, k, l_{1}$ may be called an ascending series of order $p$. It follows that $l'_{1}, k', l'$ is also an ascending series of order $p$. The relation between $l, k$ and $k, l_{1}$ is algebraic in nature.

Jacobi's first transformation allows us to express elliptic functions of a given modulus in the form of elliptic functions of a greater modulus. Its complementary transformation provides a similar relation but in form of complementary moduli and therefore helps us to express an elliptic function of a given modulus in terms of elliptic functions of a smaller modulus.

Jacobi's second transformation allows us to express elliptic functions of a given modulus in the form of elliptic functions of a smaller modulus. Its complementary transformation provides a similar relation but in form of complementary moduli and therefore helps us to express an elliptic function of a given modulus in terms of elliptic functions of a greater modulus.

From these considerations it is seen that the Jacobi's first complementary transformation is analogous to Jacobi's second transformation and Jacobi's second complementary transformation is analogous to Jacobi's first transformation.

Together both the Jacobi's transformations allow us express the multiplication formulas for elliptic functions either by first switching to a higher modulus and then back to the original modulus or by first switching to a lower modulus and back to original modulus.

In terms of the multiplier, we see that the first transformation gives $L = K/pM$, the first complementary transformation gives $L' = K'/M$, the second transformation gives $L_{1} = K/M_{1}$, the second complementary transformation gives $L'_{1} = K'/pM_{1}$.
The above relations also hold when $p$ is not prime because then the transformations can be taken as composition of the transformations corresponding to prime factors of $p$.

With these remarks we complete our presentation of the theory of transformation as developed by Jacobi in his masterpiece Fundamenta Nova. From the next post Ramanujan will take on the stage.

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