Elliptic functions and elliptic integrals

The first rung above the polylogarithms: how an eighteenth-century arc-length problem became the function theory in which the whole integral class writes its elliptic rung: the sunrise diagram, the symmetric-rate crystal walk, the first elliptic sector of the primordial sky.

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The ladder: The BootLoops integral class · Polylogarithms · Elliptic · K3 · Calabi–Yau

The name is an accident of history

Elliptic functions have almost nothing to do with ellipses. The name survives from the problem that started the subject: the arc length of an ellipse. Parametrize an ellipse with semi-axes $a > b$ as $(a\sin\theta,\, b\cos\theta)$ and the arc from the top to angle $\phi$ comes out as

$$s(\phi) \;=\; a\int_0^{\phi} \sqrt{1-k^2\sin^2\theta}\;d\theta\,, \qquad k^2 = 1-\tfrac{b^2}{a^2}\,,$$

and this integral defeats every technique in the calculus books, provably: the antiderivative cannot be written in elementary functions — a result of Liouville's theory of integration in finite terms. Eighteenth-century mathematicians hit the same wall repeatedly: the arc of the ellipse, the period of the pendulum beyond the small-angle approximation, and — most fruitfully — the arc of the lemniscateThe figure-eight curve $(x^2+y^2)^2 = x^2-y^2$; its arc length is $\int dr/\sqrt{1-r^4}$, the simplest elliptic integral., where the arc-length element is $dr/\sqrt{1-r^4}$.

Count Giulio Fagnano discovered in 1718 that the lemniscate arc can be doubled by algebra alone: given the endpoint of one arc, a rational-and-square-root formula produces the endpoint of the arc twice as long, even though neither arc length is an elementary function. Euler received Fagnano's collected papers in December 1751 — Jacobi later called that date the birthday of the theory of elliptic functions — and generalized doubling to a full addition theorem: for the lemniscatic integral $u(x)=\int_0^x dt/\sqrt{1-t^4}$,

$$u(x) + u(y) \;=\; u(z)\,, \qquad z \;=\; \frac{x\sqrt{1-y^4} + y\sqrt{1-x^4}}{1+x^2y^2}\,,$$

the exact analogue of the sine addition formula, with the quartic playing the role of $1-t^2$. An addition theorem is the signature of a good function class hiding underneath; it took another seventy years to find the class.

Legendre spent much of his career organizing the integrals themselves. His result, published in final form in his 1825–28 treatise on elliptic functions, is a complete reduction: every integral $\int R\bigl(t, \sqrt{P(t)}\bigr)\,dt$ with $R$ rational and $P$ a cubic or quartic reduces, by algebraic substitutions, to elementary functions plus three irreducible kinds. In their complete (fixed-endpoint) form:

$$K(k) = \int_0^1 \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}\,, \qquad E(k) = \int_0^1 \sqrt{\frac{1-k^2t^2}{1-t^2}}\;dt\,, \qquad \Pi(n;k) = \int_0^1 \frac{dt}{(1-nt^2)\sqrt{(1-t^2)(1-k^2t^2)}}\,.$$

The parameter $k$ is the modulus. These three functions are the periodic table of the subject at the integral level: whatever a square root of a quartic produces, it produces in $K$, $E$, $\Pi$ and elementary pieces.

Inverting the integral

The decisive step looks small. The integral $\int_0^x dt/\sqrt{1-t^2} = \arcsin x$ is awkward; its inverse, $\sin$, is one of the best-behaved functions in analysis. Abel (1827) and Jacobi (1827–29) applied the same move to the elliptic integral: define $u(\phi) = \int_0^\phi d\theta/\sqrt{1-k^2\sin^2\theta}$ and study the inverse function $\phi(u)$. The functions are Jacobi's; the notation that survives (due to Gudermann) is $\operatorname{sn}(u,k) = \sin\phi$, $\operatorname{cn}(u,k) = \cos\phi$, $\operatorname{dn}(u,k) = \sqrt{1-k^2\operatorname{sn}^2}$.

The inverse turned out to be periodic in two independent complex directions. Just as $\sin$ has period $4\cdot\arcsin(1) = 2\pi$, $\operatorname{sn}$ has real period $4K(k)$; but it also has the imaginary period $2iK'(k)$, where $K'(k) = K(\sqrt{1-k^2})$ is the complementary integral. A function meromorphic on $\mathbb{C}$ and periodic in two directions that don't lie on one real line is called doubly periodicInvariant under translation by two complex numbers $\omega_1, \omega_2$ with $\omega_2/\omega_1$ not real — so the function repeats on a two-dimensional grid, not just along a line., and that is what the term elliptic function now means: a doubly periodic meromorphic function. The integrals of the previous section are elliptic integrals; their inverses and everything built from them are elliptic functions.

Weierstrass later reorganized the subject around the periods themselves. Fix two complex numbers $\omega_1, \omega_2$ with $\tau = \omega_2/\omega_1$ in the upper half-plane, and form the lattice $\Lambda = \mathbb{Z}\omega_1 + \mathbb{Z}\omega_2$. All values of any $\Lambda$-periodic function are already taken on the fundamental parallelogram — the tile with corners $0, \omega_1, \omega_2, \omega_1+\omega_2$ — and gluing that tile's opposite edges produces a torus: the natural domain of an elliptic function is a doughnut, not the plane. On this domain Weierstrass singled out one function,

$$\wp(z) \;=\; \frac{1}{z^2} + \sum_{w\,\in\,\Lambda\setminus\{0\}} \left[\frac{1}{(z-w)^2} - \frac{1}{w^2}\right],$$

which satisfies the differential equation

$$\wp'(z)^2 \;=\; 4\,\wp(z)^3 - g_2\,\wp(z) - g_3\,, \qquad g_2 = 60\!\!\sum_{w\neq 0}\! w^{-4}, \quad g_3 = 140\!\!\sum_{w\neq 0}\! w^{-6}.$$

Read that equation as geometry: the map $z \mapsto (x,y) = (\wp(z), \wp'(z))$ identifies the torus $\mathbb{C}/\Lambda$ with the plane curve

$$y^2 \;=\; 4x^3 - g_2\,x - g_3\,,$$

an elliptic curve. This is the geometric home of the whole subject. Conversely, the periods are recovered from the curve as contour integrals $\omega_i = \oint_{\gamma_i} dx/y$ around the two independent closed loops $\gamma_1, \gamma_2$ on the torus.

Alongside $\wp$ sits the Weierstrass zeta function, defined by $\zeta'(z) = -\wp(z)$, which is not periodic but quasi-periodic — $\zeta(z+\omega_i) = \zeta(z) + \eta_i$ picks up a constant $\eta_i$, the quasi-period. Periods and quasi-periods are not independent; they satisfy the Legendre relation

$$\eta_1\,\omega_2 - \eta_2\,\omega_1 \;=\; 2\pi i\,,$$

an exact bilinear identity between transcendental numbers. It looks like a curiosity here; on the kite page its $K$-and-$E$ form, $KE' + K'E - KK' = \tfrac{\pi}{2}$, fixes a Wronskian exactly.

Finally, rescaling the lattice rescales $g_2, g_3$; the combination $j = 1728\,g_2^3/(g_2^3 - 27 g_3^2)$ is scale-invariant, and two elliptic curves are isomorphic exactly when their $j$-invariants agree. One complex number classifies the geometry. On the energy-correlator page an identity between two $j$-invariants is the whole theorem.

Periods as the bridge to physics

Now let the curve move. Take a quartic whose roots depend on an external parameter,

$$y^2 \;=\; (t-a_1)(t-a_2)(t-a_3)(t-a_4)\,, \qquad a_i = a_i(s)\,,$$

and follow the periods $\psi_i(s) = \oint_{\gamma_i} dt/y$ as $s$ varies. The complete integrals $K$ and $E$ are exactly such periods for the family $y^2 = (1-t^2)(1-k^2t^2)$, with $k$ the moving parameter. A period of a varying curve is no longer a number; it is a function of the modulus, and it satisfies a linear differential equation. Differentiate under the integral sign: each derivative in $s$ produces a new integrand, but on a fixed curve there are only two independent contour integrals, so the second derivative must be a linear combination of the function and its first derivative, up to total derivatives that integrate to zero around a closed loop. For $K$ this yields Legendre's differential equation: with $m = k^2$ and $f(m) = \tfrac{2}{\pi}K(\sqrt m)$,

$$m(1-m)\,f''(m) + (1-2m)\,f'(m) - \tfrac14\,f(m) \;=\; 0\,,$$

a second-order equation whose two solutions are the two periods, $K(k)$ and $K'(k)$. The modern name for the equation a period satisfies is the Picard–Fuchs equation of the family, and Legendre's equation is its prototype.

This is the entire mechanism by which elliptic functions enter the BootLoops integral class. A dimensionally regularized Feynman integral, viewed as a function of its kinematic invariants, satisfies linear differential equations: differentiate with respect to $p^2$ and integration-by-parts identities close the system on a finite basis of master integrals. The homogeneous part of a master's equation is governed by its maximal cutPut every internal propagator on shell at once; the integral over what survives strips away the boundary data and exposes the diagram's underlying geometry.: when the maximal cut is a period of an elliptic curve, the homogeneous operator is the Picard–Fuchs operator of that curve, and the diagram itself solves the inhomogeneous equation, with simpler diagrams as the source. Variation of parameters — the elementary differential-equations technique — then writes the answer as periods times iterated integrals over the source. Every elliptic Feynman integral on this site is built exactly this way — and the same mechanism runs in the other branches. Watson's original 1939 evaluation of the symmetric crystal-walk integral is the square of a complete elliptic integral, and the two-equal-rates case of 2001 is a product of two: the lattice branch's own elliptic rung. In cosmology, the first elliptic sector of a wavefunction coefficient closed in complete elliptic integrals with a proven algebraic dressing.

The sunrise

The canonical example, and the door through which all of this entered physics, is the two-loop equal-mass sunrise: two vertices joined by three propagators of mass $m$, with external momentum $p$,

$$J_{111}(d,\,p^2) \;=\; \int d^d l_1\, d^d l_2\; \frac{1}{\bigl(l_1^2-m^2\bigr)\bigl(l_2^2-m^2\bigr)\bigl((l_1+l_2-p)^2-m^2\bigr)}\,, \qquad t = \frac{p^2}{m^2}\,.$$

After the loop integrations the integrand carries the square root of a quartic in the remaining variable, so the natural contour threads around four branch points rather than collecting residues at poles — and a double cover of the plane branched at four points is a torus. The maximal cut of the sunrise is a period of an elliptic curve whose shape varies with $t$. Its Picard–Fuchs operator is second order and irreducible: it does not factor into first-order pieces. That irreducibility is the precise obstruction to a polylogarithmic answer, because polylogarithms are exactly the iterated integrals whose differential equations factor completely into first-order $d\log$ steps. The sunrise has resisted polylogarithmic evaluation since Sabry's calculation in 1962, and it became the proving ground for everything beyond: Laporta and Remiddi gave the full analytic treatment in 2005, Bloch and Vanhove recast the answer in the elliptic dilogarithm, and Adams and Weinzierl assembled the all-orders representation as iterated integrals of modular forms — functions of the period ratio $\tau$ with a symmetry described in the next section.

The arithmetic is unreasonably clean. The integral can only become singular at $t = 0, 1, 9, \infty$ — the pseudo-threshold $t=1$, the three-particle threshold $t=9$ — and those four points are exactly the four cuspsThe boundary points of a modular curve, where the elliptic curve degenerates; the natural anchor points for series expansions. of the modular curve of the congruence subgroupA subgroup of SL(2,ℤ) defined by congruence conditions on the matrix entries; quotienting the upper half-plane by it produces a modular curve. $\Gamma_1(6)$, and a classification theorem of Beauville picks that curve out uniquely from how the sunrise curve degenerates at those four points. The holomorphic period closes in Legendre's function:

$$\frac{\psi_1}{\pi} \;=\; \frac{4}{\pi}\,\bigl[(t-1)^3(t-9)\bigr]^{-1/4}\,K(k^2)\,,$$

with the modulus built from the roots of the sunrise quartic (the argument of $K$ here is the modulus squared, the convention of the sunrise page). Legendre's $K$, the moving curve, and the Picard–Fuchs equation all appear in that one line, inside a physical two-loop diagram. The full story, including the blind reconstruction of the answer through $\varepsilon^4$, is on the sunrise page.

Two working bases

Granted the geometry, in what functions do you actually write an elliptic Feynman integral? Two bases are in professional use, and they organize the same content two ways.

Elliptic multiple polylogarithms (eMPLs). Ordinary multiple polylogarithms are iterated integrals of $d\log$ kernels — rational one-forms with simple poles. Their elliptic analogues are iterated integrals on the torus, with kernels built from $\wp$ and its relatives (systematically, from the coefficients of the Kronecker–Eisenstein series). The mathematical theory goes back to work of Beilinson–Levin and Brown–Levin; the physics versions were developed by Broedel, Duhr, Dulat, Penante and Tancredi, and in a parallel line by Adams, Bogner and Weinzierl. eMPLs are the basis of choice when the arguments naturally live on the curve — several marked points, unequal masses — because the kernels keep track of positions on the torus explicitly.

Iterated integrals of modular forms. When the family of curves is modular — carrying the two periods around the singular points only mixes them by matrices from a congruence subgroup of $SL(2,\mathbb{Z})$, as for the sunrise on $\Gamma_1(6)$ — a change of variable from the kinematic $t$ to $\tau = \psi_2/\psi_1$ rewrites everything as iterated integrals of modular forms in $\tau$: functions on the upper half-plane satisfying $f\bigl(\tfrac{a\tau+b}{c\tau+d}\bigr)=(c\tau+d)^{k}f(\tau)$ for the matrices of the congruence subgroup, the integer $k$ being the form's weight — the simplest examples being the Eisenstein seriesExplicit $q$-series like $E_4 = 1+240\sum \sigma_3(n)q^n$ with $q=e^{2\pi i\tau}$; the basic building blocks among modular forms at each weight.. Same content, reorganized so the modular symmetry is manifest: the integration kernels become a finite-dimensional space of modular forms at each weight, every function gains a $q$-expansion around the cusp, and numerics converge geometrically. This is the convenient basis when one modulus carries all the kinematics and the family sits on a congruence curve — the situation of the equal-mass sunrise, and the basis in which BootLoops writes its solution. The depth-one building blocks are Eichler integrals — antiderivatives of modular forms, iterated $k-1$ times for weight $k$ — and the site's Eichler library exists to compute them.

Neither basis is universal. The unequal-mass kite sits on a curve attached to no congruence subgroup, so the modular dictionary is empty there, and no finite expression in eMPL iterated integrals works either — provably; the answer is instead transported along the differential equation itself from an analytically derived boundary. Knowing which representation a given curve admits — modular series, eMPL word, or transport only — is a working question on every elliptic page of this site.

Where this shows up in BootLoops

Further reading