English translation of Frobenius' and Stickelberger's "On the theory of elliptic functions"

1. Problem Statement

The paper addresses the unification and derivation of specific determinant formulas related to elliptic functions, specifically those involving the Weierstrass $\wp$-function and the $\sigma$-function. The authors aim to:

• Clarify the mathematical connections between a general formula communicated by Charles Hermite and a specialized formula used by Ludwig Kiepert to solve the multiplication problem for elliptic functions.
• Establish a rigorous limiting process to transition from general determinant expansions to these specific elliptic identities.
• Provide a unified derivation that demonstrates how these formulas arise naturally from the properties of doubly periodic functions and symmetric determinants.

2. Methodology

The authors employ a two-pronged methodological approach combining determinant theory with the analytic properties of elliptic functions.

A. The General Limiting Process for Determinants

The authors first establish a general theorem regarding determinants of convergent power series.

• Let $f_0(u), \dots, f_n(u)$ be $n+1$ convergent series.
• Consider the determinant $F = |f_\alpha(u_\beta)|$ where $u_0, \dots, u_n$ are distinct variables.
• Since $F$ is an alternating function of the variables, it can be factored as $F = G(u_0, \dots, u_n) \prod (u_\alpha - u_\beta)$, where $G$ is a symmetric function.
• The Limit: By setting $u_\beta = u + \beta h$ and letting $h \to 0$, the authors derive a relation between the limit of the quotient and a determinant of derivatives:
  $$\lim_{h \to 0} \frac{F}{\prod (u_\alpha - u_\beta)} = \frac{1}{\prod (\alpha - \beta)} \det |f_\alpha^{(\beta)}(u)|$$
  This allows the conversion of a determinant of function values at distinct points into a determinant of derivatives at a single point.

B. Analysis of a Specific Elliptic Determinant

The authors apply this logic to a specific determinant $R$ involving the logarithmic derivative of the sigma function, $\psi(u) = \frac{d}{du} \log \sigma(u)$.

• Structure: $R$ is a determinant of size $(n+2) \times (n+2)$ with a specific first row and columns involving $\psi(u_i + v_j)$.
• Periodicity and Poles: They analyze $R$ as a function of $u_0$. Since $\psi(u)$ is doubly periodic, $R$ is also doubly periodic. The poles of $R$ occur at $u_0 = -v_j$ (simple poles), and zeros occur at $u_0 = u_j$ ($j=1 \dots n$).
• Abel's Theorem Application: Using the property that the sum of zeros equals the sum of poles (modulo periods) for elliptic functions, they deduce the existence of an $(n+1)$-th zero. This allows them to express $R$ explicitly in terms of the $\sigma$-function products.

C. The Double Limit

Finally, they apply the limiting process (setting $v_\beta = \beta h$ and $h \to 0$) to the derived $\sigma$-function identity. This transforms the determinant of $\psi$ values into a determinant involving derivatives of $\psi$, which correspond to the Weierstrass $\wp$-function and its derivatives.

3. Key Contributions and Results

Result 1: The General Determinant Identity (Eq. 2)

The authors derive a fundamental identity relating a determinant of $\psi$ functions to a product of $\sigma$ functions:
$$-\det \begin{pmatrix} 0 & 1 & \dots & 1 \\ 1 & \frac{\sigma'(u_0+v_0)}{\sigma(u_0+v_0)} & \dots & \frac{\sigma'(u_0+v_n)}{\sigma(u_0+v_n)} \\ \vdots & \vdots & \ddots & \vdots \\ 1 & \frac{\sigma'(u_n+v_0)}{\sigma(u_n+v_0)} & \dots & \frac{\sigma'(u_n+v_n)}{\sigma(u_n+v_n)} \end{pmatrix} = \frac{\sigma(\sum u + \sum v) \prod \sigma(u_\alpha - u_\beta) \prod \sigma(v_\alpha - v_\beta)}{\prod \sigma(u_\alpha + v_\beta)}$$
This formula generalizes previous results by Jacobi and serves as the foundation for the subsequent derivations.

Result 2: Hermite's Formula (Eq. 3)

By applying the limiting process to the general determinant where $f_k(u) = \wp^{(k-1)}(u)$, they recover the formula communicated by Hermite. This relates a determinant of $\wp$ and its derivatives to a product of $\sigma$ functions:
$$\det \begin{pmatrix} 1 & \wp(u_0) & \dots & \wp^{(n-1)}(u_0) \\ \vdots & \vdots & \ddots & \vdots \\ 1 & \wp(u_n) & \dots & \wp^{(n-1)}(u_n) \end{pmatrix} = (-1)^n \left( \prod_{\alpha > \beta} (\alpha - \beta) \right) \frac{\sigma(\sum u_i) \prod \sigma(u_\alpha - u_\beta)}{(\prod \sigma(u_i))^{n+1}}$$

Result 3: Kiepert's Formula (Eq. 4)

By applying the limiting process a second time (collapsing the variables $u_0, \dots, u_n$ to a single variable $u$), they derive the formula used by Kiepert for the multiplication of elliptic functions:
$$\det \begin{pmatrix} \wp'(u) & \wp''(u) & \dots & \wp^{(n)}(u) \\ \wp''(u) & \wp'''(u) & \dots & \wp^{(n+1)}(u) \\ \vdots & \vdots & \ddots & \vdots \\ \wp^{(n)}(u) & \wp^{(n+1)}(u) & \dots & \wp^{(2n-1)}(u) \end{pmatrix} = (-1)^n \left( \prod (\alpha - \beta) \right)^2 \frac{\sigma((n+1)u)}{\sigma(u)^{(n+1)(n+1)}}$$

4. Significance

• Unification: The paper provides a cohesive theoretical framework that links disparate formulas by Hermite and Kiepert, showing they are not isolated results but specific cases of a broader limiting process applied to elliptic determinants.
• Rigorous Foundation: By grounding the derivation in the properties of doubly periodic functions (poles, zeros, and Abel's theorem) and standard determinant calculus, the authors provide a rigorous proof for formulas that were previously known or used heuristically.
• Advancement of Multiplication Theory: The derivation of Kiepert's formula is particularly significant as it underpins the solution to the multiplication problem for elliptic functions, a central topic in 19th-century complex analysis.
• Methodological Innovation: The specific technique of using a "difference product" limit to convert determinants of distinct arguments into determinants of derivatives at a single point is a powerful tool that likely influenced subsequent developments in the theory of special functions.

In summary, Frobenius and Stickelberger successfully demonstrated that complex identities in the theory of elliptic functions can be systematically derived through a unified limiting process, bridging the gap between general determinant theory and specific elliptic function applications.