Explicit compact form of the n-variable addition formula for the Weierstrass sigma function of a general elliptic curve (first open case n = 4; conjecturally a determinant)

The Weierstrass sigma function of a lattice satisfies, classically, addition formulae in any number of variables: the Frobenius-Stickelberger theorem expresses a certain sigma quotient as a determinant (an alternant in $\wp$ and its derivatives). Eilbeck, England and Onishi (2014) introduced a new class of $n$-variable addition formulae for the sigma function of a completely general elliptic curve $y^2+(\mu_1x+\mu_3)y=x^3+\mu_2x^2+\mu_4x+\mu_6$, built from conjugate variables $u^\star$, $u^{\star\star}$ (the three points of the curve sharing one $y$-coordinate, with $u+u^\star+u^{\star\star}=0$), proved that for every $n$ the corresponding sigma quotient equals a polynomial of weight $-(n^2-1)$ in the curve coordinates, and computed the explicit right-hand sides only for $n=2$ and $n=3$. For $n\geq4$ their naive series computation exceeded machine memory, and they state that progress should come from a more compact expression, for example a determinant; they also anticipate, without deriving, a genus-2 analogue of their 2-variable formula. The problem remains open: the only later work engaging this framework (Gürel 2026) develops a different, conjugate-variable-free family of addition theorems for the classical $\wp$-function, with explicit examples only up to three variables.

Background

Weierstrass elliptic functions and classical addition theorems. The Weierstrass $\wp$-function of a lattice $\Lambda\subset\mathbb{C}$ is the second logarithmic derivative of an odd entire function $\sigma$, the Weierstrass sigma function: $\wp(u)=-\frac{d^2}{du^2}\log\sigma(u)$; $\wp$ is doubly periodic while $\sigma$ is only quasi-periodic (under a lattice translation it acquires a non-trivial exponential factor). Classically one parametrizes the elliptic curve in Weierstrass form $y^2=4x^3-g_2x-g_3$ by $x=\wp(u)$, $y=\wp'(u)$, turning the curve into the complex torus $\mathbb{C}/\Lambda$. Addition formulae — identities expressing $\sigma$ or $\wp$ evaluated at sums of variables through their values at the individual variables — are the analytic counterpart of the group law on the curve. The oldest are the two-variable formula $-\sigma(u+v)\sigma(u-v)/(\sigma(u)^2\sigma(v)^2)=\wp(u)-\wp(v)$ and, for every $n\geq2$, the Frobenius-Stickelberger determinant: an alternant (a determinant whose successive columns evaluate a fixed list of functions at each variable) whose $j$-th row is $1,\wp(u^{(j)}),\wp'(u^{(j)}),\dots,\wp^{(n-2)}(u^{(j)})$ reproduces the quotient $\sigma(u^{(1)}+\cdots+u^{(n)})\prod_{i<j}\sigma(u^{(i)}-u^{(j)})/\prod_j\sigma(u^{(j)})^n$ up to a universal sign and factorial factor. Both formulae reflect the involution $(x,y)\mapsto(x,-y)$, that is, the map $\varphi=x$ of the curve onto the projective line $\mathbb{P}^1$ (equivalently: for fixed $x$ there are two points of the curve). Such formulae are used, for example, in the theory of integrable systems, in the inversion of elliptic integrals appearing in mechanics and general relativity, and in the arithmetic of elliptic curves.

The general curve and conjugate variables. Eilbeck-England-Onishi (2014) work with the completely general elliptic curve

$\mathscr{C}:\; f(x,y)=y^2+(\mu_1x+\mu_3)y-(x^3+\mu_2x^2+\mu_4x+\mu_6)=0,$

over the coefficient ring $\mathbb{Z}[\mu_1,\dots,\mu_6]$, together with its generalized Weierstrass sigma function $\sigma$, constructed as a universal series in $u$ and the $\mu_j$ and graded by weight (wt$(u)=1$, hence wt$(\sigma)=1$; wt$(x)=-2$, wt$(y)=-3$, wt$(\mu_j)=-j$; every formula of the paper is homogeneous for this grading); here $x(u), y(u)$ denote the coordinates of the curve point reached by integrating the holomorphic (everywhere-regular) differential $\omega=dx/(2y+\mu_1x+\mu_3)$ from the point at infinity to $u$, so that $x$ and $y$ are elliptic functions with poles only at lattice points. For a fixed value of the second coordinate $y$ the defining equation is a cubic in $x$, so a point $(x,y)$ has two further conjugate points $(x^\star,y)$ and $(x^{\star\star},y)$, and the corresponding conjugate variables $u^\star,u^{\star\star}$ satisfy $u+u^\star+u^{\star\star}=0$. On the equianharmonic curve ($g_2=0$) the conjugation becomes multiplication by a third root of unity $\zeta$, which is why the equianharmonic case admits the three-term formula $-\sigma(u+v)\sigma(u+\zeta v)\sigma(u+\zeta^2v)/(\sigma(u)^3\sigma(v)^3)=\frac12(\wp'(u)+\wp'(v))$ (Eilbeck-Matsutani-Onishi 2011). On a general curve the conjugates are genuinely new transcendental functions of $u$ with no closed form (they require solving a cubic whose coefficients involve $y(u)$), and this is the source of both the novelty and the difficulty of the new formula class.

The new class of formulae and what remains missing. Eilbeck-England-Onishi (2014, Theorem 4.2) proved that for every $n$ the quotient

$\frac{\sigma(u^{(1)}+\cdots+u^{(n)})\,\prod_{i<j}\sigma(u^{(i)}+u^{(j)\star})\,\sigma(u^{(i)}+u^{(j)\star\star})}{\prod_{j=1}^{n}\sigma(u^{(j)})^{2n+1-2j}\,\sigma(u^{(j)\star})^{j-1}\,\sigma(u^{(j)\star\star})^{j-1}}$

— an elliptic function of each variable, with poles of order $2n-1$ only at lattice points — equals a polynomial of homogeneous weight $-(n^2-1)$ in the coordinates $x(u^{(j)})$, $y(u^{(j)})$ with coefficients in $\mathbb{Q}[\mu_1,\dots,\mu_6]$. For $n=2$ the polynomial collapses to the compact expression $\frac12(\wp'(u^{(1)})+\wp'(u^{(2)}))+\frac{\mu_1}{2}(\wp(u^{(1)})-\wp(u^{(2)}))$ (their Theorem 5.1, where $\wp$ and $\wp'$ are the elliptic functions of the general curve itself, $\wp(u)=x(u)$ and $\wp'(u)=2y(u)+\mu_1x(u)+\mu_3$), and for $n=3$ it is the explicit weight-$(-8)$ polynomial of their Theorem 6.1, obtained by a computer-algebra series computation. For the classical map $\varphi=x$, the analogous $n$-variable formulae are the Frobenius-Stickelberger determinants, and such determinant representations exist for the general curve as well (determinants of monomials $x(u^{(j)})^ay(u^{(j)})^b$; Onishi's 2010 lecture notes), and in higher genus (Eilbeck-Enolskii-Previato 2003; Eilbeck-Enolskii-Matsutani-Onishi-Previato 2008; Ayano-Nakayashiki 2013). The motivating analogy — and the authors' explicit expectation — is that the new class built on the conjugate variables of the map $\varphi=y$ should likewise admit a compact, determinant-type representation; their series method cannot see it, since expanding the $n=4$ right-hand side naively exceeded the memory of their machines. The authors also record two further observations that delimit the problem: the naively generalized three-term expression built from $\sigma(u+v+w)\sigma(u+v^\star+w^{\star\star})\sigma(u+v^{\star\star}+w^\star)$ fails to be $\Lambda$-periodic on a general curve (so the quotient above is the appropriate object), and the whole construction is expected to extend to higher genus — for instance an analogue of the 2-variable Theorem 5.1 for the genus-2 curve $y^2+(\mu_1x^2+\mu_3x+\mu_5)y=x^5+\mu_2x^4+\cdots$, using the five conjugates over a fixed $y$ — though nothing of this has been worked out.

Problem Statement

Determine the explicit right-hand side of the $n$-variable addition formula of Eilbeck-England-Onishi (2014, Theorem 4.2) for the Weierstrass sigma function of the general elliptic curve $\mathscr{C}: y^2+(\mu_1x+\mu_3)y=x^3+\mu_2x^2+\mu_4x+\mu_6$, in the open cases $n\geq4$ (the first being $n=4$), and beyond the raw expansion find a compact closed representation of that right-hand side valid for general $n$.

Concretely: with conjugate variables $u^\star,u^{\star\star}$ defined by $y(u^\star)=y(u^{\star\star})=y(u)$ and $u+u^\star+u^{\star\star}=0$, give, for $n\geq4$, the explicit polynomial $P_n$ of homogeneous weight $-(n^2-1)$ in $x(u^{(j)}),y(u^{(j)})$, $j=1,\dots,n$, over $\mathbb{Q}[\mu_1,\dots,\mu_6]$, such that

$\frac{\sigma(u^{(1)}+\cdots+u^{(n)})\,\prod_{i<j}\sigma(u^{(i)}+u^{(j)\star})\,\sigma(u^{(i)}+u^{(j)\star\star})}{\prod_{j=1}^{n}\sigma(u^{(j)})^{2n+1-2j}\,\sigma(u^{(j)\star})^{j-1}\,\sigma(u^{(j)\star\star})^{j-1}}=P_n\big(x(u^{(j)}),y(u^{(j)})\big).$

A complete answer supplies at minimum a verifiable explicit $P_4$ (expanded or structured), and in full form a uniform compact expression for $P_n$ for all $n$, the conjectured shape being a determinant or comparably compact structural formula in analogy with the Frobenius-Stickelberger determinant for the $\varphi=x$ class — a form for which the authors' naive series technique fails on memory grounds alone. Any answer must respect: periodicity of both sides with respect to the lattice in each variable, poles of order $2n-1$ in each variable occurring only at lattice points, the exact weight grading, reduction to the published $n=2$ and $n=3$ polynomials (Theorems 5.1 and 6.1 of the source), and consistent behaviour under specialization of the curve, e.g. the equianharmonic specialization $\mu_1=\mu_2=\mu_4=0$ where conjugation becomes multiplication by third roots of unity.

Scope note: the source additionally states that the naive symmetric variant built from $\sigma(u+v+w)\sigma(u+v^\star+w^{\star\star})\sigma(u+v^{\star\star}+w^\star)$ is not $\Lambda$-periodic on a general curve (which is why the quotient above is the right object), and that the construction is expected to extend to higher genus, e.g. a genus-2 analogue of the 2-variable theorem on $y^2+(\mu_1x^2+\mu_3x+\mu_5)y=x^5+\cdots$. The genus-2 analogue is out of scope; the required result is the genus-1 $n$-variable explicit compact form.

References

1. J. Chris Eilbeck, Matthew England, Yoshihiro Onishi, 'Some new addition formulae for Weierstrass elliptic functions', Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470 (2014), issue 2171, article 20140051. DOI: 10.1098/rspa.2014.0051; arXiv:1207.6274. https://doi.org/10.1098/rspa.2014.0051
2. Efe Gürel, 'A recipe for obtaining algebraic addition theorems of the Weierstrass elliptic function', Integral Transforms and Special Functions (published online 9 February 2026), pp. 1-12. DOI: 10.1080/10652469.2026.2626952; arXiv:2504.12366. https://doi.org/10.1080/10652469.2026.2626952
3. Yoshihiro Onishi, 'Frobenius-Stickelberger Formulae for General Curves', lecture notes for the ICMS workshop 'The higher-genus sigma function and applications', Edinburgh, 11 October 2010. https://www.macs.hw.ac.uk/~chris/icms/Sigma/onishi.pdf
4. J. C. Eilbeck, S. Matsutani, Y. Onishi, 'Addition formulae for Abelian functions associated with specialized curves', Philosophical Transactions of the Royal Society A 369 (2011) 1245-1263. DOI: 10.1098/rsta.2010.0320. https://doi.org/10.1098/rsta.2010.0320
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6. J. C. Eilbeck, V. Z. Enolskii, E. Previato, 'On a Generalized Frobenius-Stickelberger Addition Formula', Letters in Mathematical Physics 63 (2003) 5-17. DOI: 10.1023/A:1022918717546. https://doi.org/10.1023/A:1022918717546
7. J. C. Eilbeck, V. Z. Enolskii, S. Matsutani, Y. Onishi, E. Previato, 'Addition formulae over the Jacobian pre-image of hyperelliptic Wirtinger varieties', Journal fuer die reine und angewandte Mathematik 619 (2008). DOI: 10.1515/CRELLE.2008.039. https://doi.org/10.1515/crelle.2008.039
8. Takanori Ayano, Atsushi Nakayashiki, 'On Addition Formulae for Sigma Functions of Telescopic Curves', Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 9 (2013), article 046, 14 pages. DOI: 10.3842/SIGMA.2013.046; arXiv:1303.2878. https://doi.org/10.3842/SIGMA.2013.046
9. J. C. Eilbeck, J. Gibbons, Y. Onishi, S. Yasuda, 'Theory of heat equations for sigma functions', Glasgow Mathematical Journal 67 (2025) 365-422. DOI: 10.1017/S0017089524000417; arXiv:1711.08395. https://doi.org/10.1017/S0017089524000417