A Planar Cubic Derived from the Logarithm of the Dedekind g-Function

C. A. LU¨TKEN

e first briefly recall the Weierstrass theory of Translations T ðsÞ¼s þ 1 and inversions SðsÞ¼À1=s elliptic functions. Every elliptic curve is isomor- generate the modular group C ¼ PSLð2; ZÞ, which is the set WW phic to a Weierstrass cubic of all Mo¨bius transformations with integer coefficients and unit determinant. All G with w  4 are modular forms of ðÞd}=dz 2¼ 4}3 À g } À g  CP ½Š}; d}=dz ; ð1Þ w 2 3 2 weight w, which means that they transform like tensors,  as þ b where G ¼ðcs þ dÞwG ðsÞ; 4  w 2 2Z ; ð4Þ ()w cs þ d w 1 X 1 1 }ðz; sÞ¼ þ À ð2Þ and the ring of all holomorphic modular forms is generated z2 s 2 s 2 nþms6¼0 ðz þ m þ n Þ ðm þ n Þ by G4 and G6. The Eisenstein series is the meromorphic Weierstrass function with double 3 1 du E ðsÞ¼ G ðsÞ¼DuðsÞ¼ poles, z 2 C is a point on the curve (a torus), and s 2 C is 2 p2 2 2pi ds 1 its modulus (‘‘shape’’); see Figures 1 and 2. The coeffi- derives from the logarithm of Dedekind’s eta function [3], cients are Eisenstein functions, which come in three varieties that differ only by how they are normalized: uðsÞ¼24 log gðsÞ¼log DðsÞ ; Y1 X1 wÀ1 n 2w n q gðsÞ¼q1=24 ðÞ1 À qn : EwðsÞ¼1 À ; n n¼1 Bw n¼1 1 À q ð3Þ Cusp forms are modular forms that vanish when s ! i1, GwðsÞ¼2fðwÞEwðsÞ ; 2 and the discriminant gw=2ðsÞ¼4ðw À 1Þ GwðsÞ ; ÀÁ 3 2 12 24 DðsÞ¼ g2 À 27g3 =ð2pÞ ¼ g ðsÞ where 2  w 2 2Z, q ¼ expð2pisÞ, f is Riemann’s zeta of the elliptic curve is the unique modular cusp form of function, and B2 ¼ 1=6, B4 ¼À1=30, B6 ¼ 1=42, are Ber- weight 12, up to normalization. Since noulli numbers.

1See any of the many excellent textbooks on elliptic functions, e.g., [1].

Ó 2021 The Author(s), Volume 43, Number 3, 2021 15 https://doi.org/10.1007/s00283-021-10093-7 perfect analogy with trigonometry (cf. the section on lattice functions below). Let us consider what happens to the planar Weierstrass cubic in (1) if we put the constant term back in the game,

}2ðz; sÞ¼}ðz; sÞþG2ðsÞ : ð7Þ Using the Ramanujan identities [9]

2 12DE2 ¼ E2 À E4 ;

3DE4 ¼ E2E4 À E6 ; ð8Þ 2 2DE6 ¼ E2E6 À E4 Figure 1. The two-dimensional lattice K  C is generated by s (a geometric interpretation of these may be found in the the vectors (1, 0) and ð0; sÞ spanning the fundamental lattice appendix), we find that every elliptic curve is isomorphic to K0 K cell (shaded purple). The punctured lattice s is s with the a quasimodular planar cubic origin removed. Each horizontal string of lattice points is a 2 3 2 2 one-dimensional sublattice that we will call a chain, whence ðÞd}2=dz ¼ 4} À g1} À og1}2 À o g1=6 2 2 ð9Þ Ks may be parsed as a stack of chains. 3 o 2 o2 o3 ¼ 4}2 þ u}2 þ u}2 þ u=6

CP pffiffiffiffiffiffiffiffi in 2½}2; d}2=dzŠ. Surprisingly, the coefficients in this gðs þ 1Þ¼eip=12gðsÞ; gðÀ1=sÞ¼ ÀisgðsÞ ; curious cubic are simply derivatives ou ¼ 2pidu=ds ¼ Àg1 ¼À12G2, etc., of the logarithm of Dedekind’s the Dedekind eta function transforms almost (i.e., up to g-function. phases) like a modular form of weight 1/2 on C, and it is a Thus, every elliptic curve is ‘‘contained’’ in the logarithm modular form of weight 1/2 on the metaplectic double of the Dedekind g-function (or equivalently, the logarithm cover of C. It has many subtle and surprising connections of the discriminant), since it is completely determined by to other areas of mathematics, including number theory, the first three derivatives of log g (or log D) evaluated at the topology, index theory of elliptic operators, algebraic point s on the modular curve that gives the shape of the geometry, and gauge theory [2]. elliptic curve (up to modular transformations). What follows is a more or less self-contained introduc- Quasimodular Planar Cubics tion to elliptic functions, which will enable us to derive (9) directly from first principles, i.e., without the aid of Because the sum defining G in (3) is not uniformly con- 2 Weierstrass theory. The presentation is not the conven- vergent, it does not transform like a modular form as in (4),  tional one found in textbooks, but it is arguably a more as þ b intuitive approach that may serve as a precursor to the G ¼ðcs þ dÞ2G ðsÞÀ2picðcs þ dÞ : ð5Þ 2 cs þ d 2 classical theory.2 It is also better adapted to the task of studying pinched tori, i.e., the large complex structure limit s ! i1, where the tension between holomorphy and This had to be the case, since no modular form on C of automorphy (modular symmetry) that infects modular weight less than four exists. Rather, iG =p transforms like a 2 mathematics becomes acute. This should not be sup- connection on the modular curve X, which is Cþ=C com- pressed, as is usually done, but confronted head on from pactified by gluing the rational numbers to the upper half- the beginning so that we can see how it fits into the story. plane, called a quasimodular form of weight two. It is Anomalous symmetries play a fundamental role in quan- implicit in the work of Weierstrass, as will be explained tum field theory, and we shall see that quasimodular below, but quasimodular forms were first studied system- symmetries and the associated holomorphic anomaly are atically by Ramanujan [9], who called them mock theta equally important in understanding properties of singular functions, and a decade later by Hecke [6]. geometries. This lack of modularity is why Weierstrass removes the constant term G2 from the Laurent expansion of the would- be modular } on a disk punctured at the origin, leaving Elliptic Functions X1 1 2k Periodic (circle) functions are called trigonometric func- }ðz; sÞ¼ þ ð2k þ 1ÞG2kþ2ðsÞz : ð6Þ z2 tions. A product of two circles is a torus, which after a point k¼1 of origin has been chosen is called an elliptic curve. If we do not impose any constraints, it is way too easy to make The price he pays for this is that his sigma and zeta func- doubly periodic functions: the product P(x)Q(y) of any two tions have parts containing G2 that spoil an otherwise periodic functions P and Q is doubly periodic, frequently

2This presentation is, at least in spirit, much closer to Eisenstein’s original work than to the subsequent (and now universally adopted) approach taken by Weierstrass [4, 10].

16 THE MATHEMATICAL INTELLIGENCER finite, but never (by Liouville’s theorem) holomorphic (complex analytic), except for constants; cf. Figure 2(a). The useful compromise is to consider meromorphic doubly periodic (toroidal) functions, which are called elliptic functions. The minimum amount of divergence is to have a double pole (which may be split into two simple poles) per lattice cell; cf. Figure 1. The prototypical elliptic function is the Weierstrass }-function defined in (2), which is plotted on a square torus (s ¼ i) in Figure 2(b), and on a rectangular torus with s ¼ 2i in Figure 2(c). If the real part R} or imaginary part I} is plotted instead of (as here) j}j, then it is obvious that it has double poles; cf. Figures 5 and 6 . We can regard both periodic and doubly periodic functions as lattice functions, i.e., lattice sums that are manifestly periodic in one or two directions. Furthermore, we shall view a two-dimensional (2D) lattice as a stack of one-dimensional (1D) lattices, each of which is a string of lattice points parallel to the real line (cf. Figures 1 and 2 ) that we call a chain. We can dissect 2D sums by doing one chain at a time, and we therefore suspect that elliptic and trigonometric functions are close cousins. That this is indeed the case is most easily seen by constructing both as lattice sums. For example, we shall soon see that the closest cousin to }ðzÞ is p2 csc2 pz. A chain is a very specific horizontal linear string of points that stay together when the complex structure s of the torus is changed. This changes how far apart the chains are, but they do not change shape, and it is therefore nat- ural to treat them as building blocks of the 2D lattice; cf. Figures 1, 2, and 6 . This carries over to chain functions, which are 1D lattice sums of rational functions that define trigonometric func- tions. When these functions are complexified, the chain Figure 2. Doubly periodic functions are rarely elliptic. (a) The functions are glued together by the complex structure, but real doubly periodic function sin x sin y, with some of the they retain their identity as building blocks of elliptic functions. graph removed to reveal the checkerboard symmetry. This is This parsing of a 2D lattice as a stack of chains highlights not an elliptic function. The Weierstrass function }ðz; sÞ is the similarities between 1D and 2D lattice functions, and it elliptic, with one double pole on each lattice cell: (b) j}ðz; iÞj is the main pedagogical device used here to explain that (square torus), (c) j}ðz; 2iÞj (rectangular torus). Notice how elliptic lattice functions (rather than Weierstrass functions) strings of lattice points move apart as Is increases, eventually are the closest relatives of trigonometric functions. becoming individual 1D lattices that we here call chains. Our convention is that the lattice Ks is 1D with period þ s if s is a positive real number, i.e., Ks ¼ sZ when s 2 R . want to have a unified treatment of multiperiodic functions The lattice Ks is 2D with basis ð1; sÞ if s is not real that exposes cross-dimensional similarities, which can then (Is 6¼ 0), and we can without loss of generality parame- be exploited in the construction of elliptic functions. Our terize all such lattices by the upper complex half-plane aim is to be as concrete and explicit as possible, using only Cþ ¼fs 2 C j Is [ 0g, elementary analysis and checking results by plotting them. Subtle convergence questions (convergence, absolute Z Z Cþ Z Ks ¼ Â s ¼fm þ ns j s 2 ; m; n 2 g : convergence, or uniform convergence) are effectively avoided by exploiting well-known analytic properties of Doubly periodic functions are obtained by tracing fractions special (zeta and polygamma) functions. Z (rational functions) over a 2D lattice Ks62R. The lattice is K1 ¼ . The choice of origin on the lattice is arbitrary, but an origin is required in order to have a group structure on the lattice (elliptic curves are abelian Chain Functions groups). Functions that are manifestly periodic can be We first consider 1D periodic functions from the lattice obtained by cloning a rational function on each link. The point of view, preparing a template onto which we later simplest seed function that has any chance of giving a Àn can graft doubly periodic functions. In other words, we convergent sum is x for sufficiently large n, which is

Ó 2021 The Author(s), Volume 43, Number 3, 2021 17 copied onto each link of the lattice K1 ¼ Z by translation. invariant under translations T ðxÞ¼x þ 1, since we can The trace over the lattice is a superposition of these link relabel all lattice points, but note that this works only functions, i.e., the lattice sum because the lattice is infinite in both directions. No points X ‘‘drop off the end’’ when shifted left or right, because there 1 R Z pnðxÞ¼ n ; x 2 n ; 10 are no end points. If the lattice sum is regularized by ðx þ mÞ ð Þ m2Z chopping off all but a finite piece in the middle, as is often which has poles of order n when x 2 Z. Since the sum is done, then this symmetry is brutally violated. Since our absolutely convergent when n  2, it can be differentiated motivation for tracing over infinite lattices is to build term by term, giving the recurrence relation (d/dx is manifestly periodic functions, it is not very appealing to abbreviated as ox) immediately destroy this circle symmetry by cutting off the sum at both ends in order to make it finite, even if peri- 1 1 nÀ1 pnþ1 ¼À oxpn ¼ ðÞÀox p2; n  1 : odicity is eventually restored. We prefer a more benign n n! scheme that respects symmetries. The finiteness of analytic schemes is not as explicit, but So we need to compute only p2, but it is instructive to they are continuous and do minimal harm to the geometric calculate all the pn directly, using a method that is tailor- structure of the theory. In particular, they respect all sym- made for the analytic regularization that will follow. metries as far as possible.4 Since symmetry is the DNA of For n  2, the Riemann–Hurwitz zeta function is any quantum theory, analytic schemes are usually preferred X1 n 1 ðÀ1Þ in physics. They are essentially variations on Riemann’s fðn; xÞ¼ ¼ wðnÀ1ÞðxÞ ; analytic continuation of the series that defines his zeta ðx þ mÞn CðnÞ m¼0 function: embed the sum into a larger analytic family, using ðnÞ onþ1 a reflection formula if possible, and then define the value of where w ðxÞ¼ x log CðxÞ (n  0) is the polygamma function. Equation (10) can therefore be written as the suspicious series to be the limit of a sequence of well- behaved family members. Here we need only the (gener- n pnðxÞ¼fðn; xÞþðÀ1Þ fðn; 1 À xÞ alized) Riemann zeta function. hi 1 Sometimes it is not possible to regulate a function ¼ wðnÀ1Þð1 À xÞþðÀ1ÞnwðnÀ1ÞðxÞ : CðnÞ (theory) without harming its geometric structure, i.e., without breaking some symmetry. This generates so-called Using that the polygamma function satisfies the reflection anomalies that carry subtle and useful information about formula the geometric structure. In the theory of elliptic functions, n ðnÞ ðnÞ on ðÀ1Þ w ð1 À xÞ¼w ðxÞþp x cot px; n  0 ; ð11Þ they give an explicit parameterization of the conflict between holomorphy and automorphy suffered by modu- we obtain lar forms of low weight. If it is impossible to maintain both p o nÀ1 a holomorphic structure and modular symmetry, then the pnðxÞ¼ ðÞÀ x cot px; n  2 : ð12Þ CðnÞ modular anomaly explicitly exhibits the problem, and it can be used to construct functions that respect one or the other As expected, all these periodic functions are trigonometric. of these geometric structures, but not both simultaneously. We would like to extend this result to the case n ¼ 1, This will play an important part in our discussion, since but it is not even obvious that p is well defined, since the we are particularly interested in the boundary of moduli 1 5 zeta function has its only pole at n ¼ 1. Whenever we space, where this problem becomes acute. In order to encounter a sum of dubious convergence, we must regu- analyze the so-called nodal limit, in which a torus is late it.3 Regularization is a procedure that extracts a finite pinched, we need a holomorphic function that transforms part from a (potentially) divergent series. As long as the anomalously under modular transformations. It is nearly a procedure satisfies the Hardy axioms [5], the result should weight-two form, and it emerges naturally from lattice sums be independent of the chosen regularization scheme. The that are convergent, but not uniformly convergent. This most popular schemes (in both physics and mathematics) subtle aspect of infinite series is respected by our analytic are cutoff regularization and analytic regularization. regulator, and the required anomaly appears automatically. A cutoff scheme makes everything explicitly finite at This is a good reason to use an analytic regulator. If a cutoff every step until the limit is taken at the end. However, a scheme produces an anomaly, how can we be absolutely cutoff is a discrete regulator that typically breaks every sure that this is not an artifact of the explicit symmetry- breaking introduced by the regularization scheme? How do symmetry in sight. For example, the lattice K1 ¼ Z is

3Regularizing a quantum field theory, which usually is rife with infinities, is a prerequisite for the infamous procedure called renormalization. This extracts the physical content from apparently ill-defined expressions and may be regarded as a very adult version of our discussion of p1. 4Classical symmetries are sometimes broken by quantum fluctuations. If this anomalous symmetry is local (gauge invariance), then this is fatal, and the quantum field theory is discarded, but anomalous global symmetries may be tolerable (since global symmetries typically are broken in the real world), and even useful. 5The original physical motivation is that this is the weak coupling limit of a particular effective quantum field theory that captures universal properties of quantum Hall systems. This theory is a nonlinear sigma model with a toroidal target space, which degenerates to a nodal curve in the perturbative domain [7, 8]. Conventional quantum perturbation theory is a nonmodular asymptotic expansion adapted to the singular geometry. The broken (or hidden) modular symmetry is presumably related to the holomorphic anomaly.

18 THE MATHEMATICAL INTELLIGENCER 0 we know that other symmetry-violating schemes do not functions GwðsÞ on Ks in Weierstrass theory. Not surpris- produce different anomalies? The anomaly would then not ingly, from (3) we see that cw is in fact the leading (s- be canonical, and would not carry any invariant geometric independent) term in the q-expansion of Gw, so we are on n or topological information. the right track. Since pnðxÞ¼ðÀ1Þ pnðÀxÞ, these periodic So, since the zeta function can be analytically continued lattice functions are parity eigenstates, and we have found to C 3 s 6¼ 1, we investigate the analytic continuation the two complete families of odd/even functions shown in X Figure 3. 1 s psðzÞ¼ ¼ fðs; zÞþðÀ1Þ fðs; 1 À zÞ Equation (13) gives a hierarchy of algebraic and differ- ðz þ mÞs m2Z ential equations: of p in the limit as s ! 1. If p is well defined, the poles of n 1 p ¼Àlog sin px ; the two zeta functions must cancel, and they do: 0 0 p1 ¼Àp0 ¼ p cot px ; p1ðxÞ¼lim psðxÞ 0 2 2 2 2 s!1p2 ¼Àp1 ¼ p1 þ p ¼ p csc px ; 1 1 1 1 lim wðsÀ1Þ x wðsÀ1Þ 1 x 0 00 3 2 ¼ À ð ÞÀ þ ð À Þ p3 ¼À p2 ¼ p1 ¼ p1p2 ¼ p cot px csc px ; s!1 s À 1 s À 1 2 2  ¼ wð1 À xÞÀwðxÞ¼p cot px ; 1 1 2p2 p ¼À p0 ¼ p00 ¼ p p À ; 4 3 3 6 2 2 2 3  2 where use has been made of the reflection property of the 1 0 p p5 ¼À p4 ¼ p1p2 p2 À ; digamma function wðxÞ¼wð0ÞðxÞ recorded in (11). 4 3 Equation (12) can therefore be extended to include the ; case n ¼ 1, since this also gives p1ðxÞ¼p cot px. Finally, 0 o where p0 ¼ xp0 ¼ dp0=dx, etc. The two expressions for p4 since p1ðxÞ¼Àoxp0ðxÞ with p0ðxÞ¼Àlog sin px, we can give a second-order differential equation for p2, while a obtain all pn from the ‘‘potential’’ p0 by derivation: first-order equation is obtained by squaring p3 and using n 2 2 ðÀ1Þ that p ¼ p2 À p , p ðxÞ¼ onp ðxÞ; n  1 : ð13Þ 1 n CðnÞ x 0 ÀÁ 2 1 0 2 2 2 2 2 p3 ¼ p2ðxÞ ¼ p1p2 ¼ p2 p2 À p : As a final check, we expand the pnðxÞ defined by (10) for 4 jxj \ 1, This is a cubic equation in (renormalized) chain functions,  2 3 6 1 which with X; Y p =p ; p =p is the cubic 1 X n þ k À 1 ð Þ¼ð 2 3 Þ p ðxÞ¼ þ c ðÀxÞk ; n xn nþk 2 2 0 2 R2 k¼0 n À 1 Y ¼ X ðX À 1Þ¼ðX Þ =4  : ð14Þ X 1 c ; K0 Z : nþk ¼ nþk 1 ¼ nf0g 0 m Although this looks like a planar cubic (a nodal curve in the m2K1 complex projective plane CP2), as a real equation it is We observe that odd coefficients (n k 2r 1) vanish, þ ¼ þ equivalent to the conventional quadratic equation u2 þ while even coefficients (n þ k ¼ 2r) are zeta functions, v2 ¼ 1 that embeds the circle in the real plane R2ðu; vÞ.At X 1 the moment, this just looks like a very clumsy way of c ¼ ¼ 2fð2rÞ : 2r m2r writing trigonometric identities, but when these functions m2K0 1 are promoted to elliptic functions, they will become non- Using that fð0Þ¼À1=2, we find that p1ðxÞ must be the trivial statements. The corresponding elliptic equation cotangent function, embeds a pinched torus in the complex projective X1 X1 plane CP2. 2rÀ1 2rÀ1 p1ðxÞ¼À c2r x ¼À2 fð2rÞ x ¼ p cot pxv ; r¼0 r¼0 since it is well known that this is a generating function for Stereography of Circles zeta functions. Notice also that We give a geometric interpretation of real chain functions and the algebraic equations they satisfy that will be useful X1 1 1 later. Consider first the projective representation of the p ðxÞ¼ þ ð2k þ 1Þc x2k ¼ þ c þ 3c x2 þÁÁÁ : 2 x2 2kþ2 x2 2 4 2 R2 k¼0 circle Y ¼ Xð1 À XÞ shown in Figure 4. Every point (X, Y) on this circle, except the origin, is projected onto the line X ¼ 1 by finding the point (1, c) on the line where a Comparison with (6) and (7) shows that the 1D lattice ray from the origin through (X, Y) intersects it.7 If the ray 0 functions cw ¼ 2fðwÞ on K1 are analogous to the 2D lattice makes an angle u with the Y-axis, then c ¼ cot u, and the

6 0 3 2 2 Notice that the ‘‘Weierstrassian’’ choice YW ¼ p2=p ¼À2Y would give YW ¼ 4X ðX À 1Þ. The analogous construction in the elliptic case explains the factor of 4 in (1). 7In Figure 4, this line is called a screen, by which we mean a one- or two-dimensional manifold onto which another shape is projected. In two dimensions, discussed in the section on pinched nodal cubics below, the analogous ‘‘projection screen’’ is a 2D plane, for which the terminology is more appropriate.

Ó 2021 The Author(s), Volume 43, Number 3, 2021 19 Figure 4. Stereography of real quadrics and cubics. Simulta- neous parametric plot of the quadric Y 2 ¼ Xð1 À XÞ (orange) and the cubic Y 2 ¼ X2ðX À 1Þ (purple). Solid curves have c 2À½Š1=2; 2 , while the dashed parts of the curves are obtained for other values of c, i.e., c 2 hÀ1; À1=2i[h2; 1i. A pair of points on the quadric (orange) and cubic (purple) parameterized by c both project to the same point (1, c) (black) on the line x ¼ 1, here called a (projection) screen. The point (0, 0) on the circle projects to i1 on the screen and on the cubic, which are compactified by adding this point.

Note that both the projection and its inverse are rational transformations. The line X ¼ 1 is therefore birational to the circle almost everywhere. The exception is the point ðX; Y Þ¼ð0; 0Þ, where the transformations are singular. In order that this point also have an image on the real line, a single new point at infinity is added to the screen X ¼ 1, which is thereby closed at infinity to form a topological circle. In short, the full circle is birational to the compact- ified line, and no structure has been lost by this procedure. Consider next the real cubic given by (14), which is also plotted in Figure 4. Since it is built from periodic (circle) functions, we expect it to be related to a circle. It does not look much like a circle, but neither did the line X ¼ 1 in the Figure 3. The periodic lattice functions split into two families. example above. In order to see what is going on, we also (a) pn is odd for odd n  1; p1 fits snugly into this family, as it project the cubic onto this line. Using again y ¼ cx, we get should if the lattice trace has been properly regularized. (b) pn P2 is even for even n  0. (c) Magnification of (b) showing how ðX; Y Þ À! ð1; cÞ¼ð1; Y =XÞ ; the potential p0 fits into the even family. PÀ1 ÀÁ ÀÁ ð1; cÞ À!2 ðX; Y Þ¼ 1 þ c2 ð1; cÞ¼ csc2 u; cot u csc2 u : projection is given by

P1 With the identification u $ px, we see that we have ðX; Y Þ À!ð1; cÞ¼ð1; Y =XÞ¼ð1; cot uÞ : 2 2 2 recovered the functions p2 ¼ p X ¼ p csc u and p3 ¼ The inverse projection is obtained by inserting Y ¼ cX into p3Y ¼ p3 cot u csc2 u previously obtained from lattice the quadric Y 2 ¼ Xð1 À XÞ, sums. In other words, the lattice functions are a kind of  À1 stereographic coordinates for this real cubic curve. A sim- P1 1 sin 2u 2 ilar result in the complex case will illuminate the nodal ð1; cÞ À!ðX; Y Þ¼ 2 ð1; cÞ¼ sin u; : 1 þ c 2 limit.

20 THE MATHEMATICAL INTELLIGENCER Since both P1 and P2 are birational transformations, so are compositions of these maps: X 2 þ Y 2 PÀ1  P ðX; Y Þ¼ ðX; Y Þ 2 1 X 3 is a rational map of the quadric to the cubic, and X PÀ1  P ðX; Y Þ¼ ðX; Y Þ 1 2 X 2 þ Y 2 is a rational map of the cubic to the quadric. This shows that this cubic is a birational projective representation of the circle. In short, by gluing in the point at infinity, the cubic is compactified and given the topology of a circle. When we Figure 5. The Weierstrass function R}ðz; sÞ (green) plotted on consider tori, this will become more explicit. Taking the top of the stack R½}2ðz; sÞÀG2ðsފ (red) of complex trigono- two simplest slices of the torus gives two orthogonal circles 2 2 metric chains p2ðz þ msÞ¼p csc pðz þ msÞ, with m ¼À2; that can be used to generate the torus, and in the projective À1; 0; 1; 2 and s ¼ i. The quilt appears because of tiny random picture, one of them typically looks like the cubic curve in numerical differences between the two functions. Compare also Figure 4. Figure 6, where the chain stacking is investigated in more detail. The manner in which a real cubic can faithfully represent a circle is closely related to how a pinched torus can represent a sphere. The analogous result in 2D is that a planar nodal cubic (a degenerate complex cubic n ¼ 0; 1; 2, and more generally to a meromorphic function } ðz; sÞ, using analytic continuation. equation in the complex projective plane CP2)isstere- s ographically birational to the complex line CP S2,i.e., The recurrence relation is as before, 1 ’  a sphere. 1 d 1 d n } ðz; sÞ¼À } ðz; sÞ¼ À } ðz; sÞ ; ð15Þ In summary, chain sums give a simple family of periodic nþ1 n dz n n! dz 1 functions that may be derived from the potential function where }1 is given by a q-expansion, p0ðxÞ¼Àlog sin px (by derivation), which is perfectly adapted to (real) stereography. The main virtue of this }1ðz; sÞ¼p1ðzÞþS1ðz; sÞ ; approach to trigonometry is that it immediately generalizes X1 to stacks of chains, i.e., 2D lattices, i.e., elliptic functions. In S1ðz; sÞ¼ ½Šp1ðz À msÞþp1ðz þ msÞ doing so, the large complex structure limit of the torus, m¼1  where it is pinched down to a topological sphere, is illu- X1 1 1 ¼ 2pi À ; minated, thereby clarifying the somewhat mysterious qmu À 1 qm=u À 1 modular/holomorphic anomaly that gives rise to quasi- m¼1 modular (mock modular) forms that have recently with q ¼ expð2pisÞ and uðzÞ¼expð2pizÞ. In the last line, resurfaced in various contexts. we used that p1ðzÞ¼p cot pz, derived with great care above in the section on chain functions. Feeding this back into the recurrence relation gives  Lattice Functions 1 d n } ðz; sÞ¼p ðzÞþ À2piu S ðz; sÞ : Elliptic functions can be constructed from the simplest nþ1 nþ1 n! du 1 lattice sums in 2D, regularized using 2D ‘‘zeta functions,’’ in close analogy with the 1D case. Note, however, that while a For jzj \ 1 we can expand }n in a power series: 2D lattice is exactly the same as a stack of 1D lattices or  1 X1 n À 1 þ m chains (cf. Figure 1), elliptic (2D) lattice functions are not } ðz; sÞ¼ þ G ðsÞ ðÀzÞm ; n zn nþm simply stacks of real chain functions (cf. the section on m¼0 n À 1 X X X ð16Þ elliptic functions above). When chain functions are com- 1 1 GnðsÞ¼ ¼ n ; plexified, they become glued together by the complex 0 w 0 ðk þ msÞ w2K m2K k2K1 structure of the 2D manifold to form meromorphic func- s 1 tions. The sums where the Gn by definition are the expansion coefficients of } . Since G ¼ðÀ1ÞnG , these coefficients vanish for X 1 1 n n } ðz; sÞ¼ ; n  3 ; odd n. For even n, we evaluate them using f-functions [1], n ðz þ wÞn w2Ks n X1 ð2piÞ knÀ1qk G ðsÞ¼2fðnÞþ2 : n C n 1 qk ð Þ k¼1 À where Ks is the lattice shown in Figure 1, are explicitly doubly periodic and meromorphic (elliptic) when they are well defined, i.e., absolutely convergent (n  3). We wish The real part of the Weierstrass function is very similar to to expand this family of functions to the missing cases the 1D lattice function p2ðxÞÀp. They are by construction

Ó 2021 The Author(s), Volume 43, Number 3, 2021 21 close relatives, but they are not the same, because the other Figure 6 shows why the Weierstrass function }ðz; sÞ chains contribute a small amount also for real arguments. may be regarded as a stack }2ðz; sÞ of trigonometric chain Consider, for example, functions p2ðz þ msÞ (compare also Figure 5). We see how p2ð1=2ÞÀp ¼ pðp À 1Þ¼6:72801, which should be com- the Weierstrass function } reduces to a single trigonometric pared to chain function p2 csc2 pz shifted by the anomaly p2=3 in the large complex structure limit s ! i1, where the ‘‘bagel’’ is }ð1=2; iÞ¼} ð1=2; iÞÀG ðiÞ 2 X 2 pinched off to a ‘‘bun’’ (topologically). In other words, the 2 2 2 ¼ p sech pm À p ¼ p1 ¼ 6:87519...; planar cubic degenerates to a nodal curve that is birational m2Z 2 to a 2-sphere: CP2½3ŠÀ!CP1 ’ S , as will be discussed in where more detail in the section on pinched nodal cubics. In short, it is the 2D lattice functions }n that enjoy a Cð1=4Þ2 p p p : ... = flawless analogy to trigonometry. The lattice function }2 ¼ 1 ¼ G ¼ 3=2 ¼ 2 6220575542  21 8 ð2pÞ }ðz; sÞþG2ðsÞ differs from the Weierstrass function } by the modular anomaly G2, and this accounts for all the is the lemniscate constant,8 G is Gauss’s constant, and awkward bits in the contrived analogy between Weierstrass theory and trigonometry. This vindicates our starting point, G2ðiÞ¼p has been used. In order to banish any lingering doubt about the veracity of the stack picture of elliptic which was to fully exploit the obvious similarities between lattice functions in one and two dimensions (periodic and functions, both }ðz; iÞ and }2ðz; iÞÀp have been plotted on top of each other in Figure 5. Including only the four doubly periodic functions). chains m ¼À2; À1; 1; 2 nearest to the m ¼ 0 chain is suf- Finally, we derive a differential equation satisfied by the ficient to make the two functions numerically lattice functions, using the series expansion in (16). Since 2 indistinguishable. }2 has double poles, the leading-order pole of ðÞd}2=dz is 6 3 We can bring the chain potential p0ðxÞ¼Àlog sinðpxÞ 4=z . This can be eliminated by subtracting 4}2, leaving a 2 into a form suitable for generalization to two dimensions by pole of order four, which is eliminated by adding 12G2}2, exponentiating it and using Euler’s product formula for the which leaves a pole of order two, which is canceled by r } 2 sine function. This leads us to define K and 0 by subtracting 12ðG2 À 5G4Þ}2. The series expansion of this  pole-free function is Y z2 r ðz; sÞ¼eÀ}0ðz;sÞ ¼ pz 1 À ! sin pz ; ÀÁ K w2 2 3 2 2 K0 ðÞd}2=dz À4}2 þ 12G2}2 À 12 G2 À 5G4 }2 w2 s ÀÁ17 2 3 2 ð Þ C 0 ¼ 415G2 G4 À G2 À 35G6 þOðz Þ ; where z 2 , w take values in a 2D punctured lattice Ks (cf. Figure 1), and the chain limit s ! i1 is shown. For com- 2 parison with Weierstrass, we also relabel }1 as fK, and }2 where the remainder Oðz Þ is a regular function. Since the as }K: only holomorphic function on a torus is constant (by 2 d Liouville’s theorem), we have Oðz Þ¼0. This may also be fKðz; sÞ¼ log rKðz; sÞ¼}1ðz; sÞ!p cot pz ; verified order by order in the series expansion by explicit dz calculation. Finally, using the Ramanujan identities from d 2 2 }Kðz; sÞ¼À fKðz; sÞ¼}2ðz; sÞ!p csc pz : (8), derived from the geometry of the modular curve X in dz the appendix, we obtain (9). For reasons that soon will be These should be compared with the corresponding obvious, we shall sometimes refer to this as the stereo- Weierstrass functions: graphic cubic. The Weierstrass cubic in (1) is recovered by 2 2 2 z g2ðsÞ=2À}0ðz;sÞ p z =6 substituting }2 ¼ } þ G2 in (17). rW ðz; sÞ¼e ! e sin pz; d f ðz; sÞ¼ log r ðz; sÞ W dz W 2 Papillon Plots ¼ }1ðz; sÞþzg2ðsÞ!p cot pz þ p z=3 ; We wish to compare the Weierstrass cubic to the ‘‘stereo- d }ðz; sÞ¼À fW ðz; sÞ graphic’’ equation solved by the lattice functions }2 and dz 0 2 2 2 }2 ¼ d}2=dz ¼À2}3. ¼ }2ðz; sÞÀg2ðsÞ!p csc pz À p =3 ; While the complexification and projectivization of geometry is the key to modern mathematics, it is not easy to where g2 is Weierstrass’s eta function, not to be confused visualize. Even the simple cubic ‘‘curve’’ in the projective with Dedekind’s eta function. Comparison with (7) shows ‘‘plane’’ CP2 is really (using real numbers) the intersection that g2ðsÞ¼G2ðsÞ, which may also be verified by direct of two 3D hypersurfaces in a 4D projective space. We calculation. therefore introduce a graphical device that goes some way

8 The lemniscate constant p1 (in our nonstandard notation) has gone out of fashion, but it is a fundamental transcendental constant that plays the same role for doubly periodic functions as p does for periodic functions. The lemniscate in question appears if you slice a bagel with the knife tangential to the hole. Notice that the rational approximation p1  21=8 is surprisingly similar to Archimedes’s famous result p  22=7 (sometimes called the engineering value, because it is sufficiently accurate for many practical applications), both good to about one part per thousand.

22 THE MATHEMATICAL INTELLIGENCER Figure 7. Portrait (papillon plot) of the simple cubic family y2 ¼ x3 À ax2 þða À 1Þx, with increasing real values of a out of the plane of the paper.

y2 À y2 ¼ CxðÞÀðÞa þ 3a x x2 ; 1 2 ÂÃ1 2 ÀÁ3 1 2 2 2 2y1y2 ¼ a1 þ 2a2x1 þ a3 3x1 À x2 x2 : We obtain maximal simplification by choosing the first section to be x2 ¼ 0. Then either y1 ¼ 0ory2 ¼ 0, giving two orthogonal sections:

2 x2 ¼ 0 ¼ y2)y1 ¼ CxðÞ1 ; 2 x2 ¼ 0 ¼ y1)y2 ¼ÀCxðÞ1 : The 2D papillon (plot of one family member) is obtained by flattening these orthogonal sections into one 2D dia- gram composed of four branches (parametric plots): nohipffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2ðxÞ¼ x; Æ ÆCðxÞ jx 2 ðÞxÀ; xþ : This rotation does not do justice to the geometry of these Figure 6. The Weierstrass function }ðz; sÞ is the stack } ðz; sÞ 2 surfaces, but it does preserve the homotopy, i.e., it respects of complex trigonometric chain functions p ðz þ msÞ¼ 2 the topology of the 1-cycles found by sectioning the cubic. 2 2 Z p csc pðz þ msÞ (m 2 ), except for the modular anomaly A 3D papillon, or family plot, is a two-parameter plot G2ðsÞ, which has been removed from } because it is only nohipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi quasimodular (compare also Figure 5). (a) The m ¼ 0 P3ðx; tÞ¼ t; x; Æ ÆCðx; tÞ jx 2 ðÞxÀ; xþ ; t 2 ðÞtÀ; tþ : 2 2 chain shifted by G2ðiÞ¼p: Rp2ðzÞÀp ¼ p R csc pz À p. 2 2 (b) Ip2ðzÞ¼p I csc pz. (c) R}ðz; iÞ. (d) I}ðz; iÞ. (e) R}ðz; 2iÞ. (f) R}ðz; 4iÞRp2ðzÞÀp, since G2ðs ! i1Þ ! The real parameter t is often chosen to be one of the p2=3  p [compare (a)]. coefficients ai, and the other coefficients may also depend parametrically on t, i.e., an ¼ anðtÞ. As an example of this graphical device, a particularly simple family of cubics is exhibited as a papillon plot in Figure 7. toward visualizing these objects. Clearly, there is no way to The top row of Figure 8 shows how the 2D papillon plot retain all information about the cubic in a plot, but a judi- develops as the torus is pinched by shrinking a short cycle. cious choice of ‘‘sections’’ or ‘‘slices’’ allows us to image the The pictures in the middle row are cartoons obtained by skeleton, i.e., the generating cycles. We consider ‘‘family ‘‘closing the circles at infinity’’ (where they all meet). The plots’’ that we shall call papillon plots. The main idea is to resulting plot is a flattened picture of the cycles exposed by ‘‘flatten’’ each pair of cycles onto a plane, and then stack cutting the shapes in the bottom row. Similarly, Figure 9 these planes together as one of the parameters of the cubic shows how the 2D papillon plot develops as the torus is is changing. They are constructed as follows for the general pinched by shrinking a long cycle. pure (no mixed terms) cubic Y 2 C X a a X a X 2 a X 3; X; Y C : ¼ ð Þ¼ 0 þ 1 þ 2 þ 3 2 Pinched (Nodal) Cubics If the coefficients an are complex, then even this simple One virtue of the lattice basis is that it allows us to examine curve is too complicated, so we restrict attention to an 2 R. in a very explicit way how the torus degenerates to a nodal With X ¼ x1 þ ix2 and Y ¼ y1 þ iy2 (x1; x2; y1; y2 2 R), this cubic, which by a stereographic transformation is bira- is equivalent to the two real equations tionally equivalent to a sphere.

Ó 2021 The Author(s), Volume 43, Number 3, 2021 23 Figure 9. Nodal cubic: shrinking the long cycle. Figure 8. Nodal cubic: shrinking the short cycle.

Consider first the family of rectangular tori parameter- ized by s ¼ it, where t is a positive real number. Since qðitÞ¼expðÀ2ptÞ is real, so are all the coefficients of the stereographic cubic in (9) [cf. (3)], and we can construct the papillon plot shown in Figure 10. This plot shows very clearly how a node emerges as t grows larger than 1 and completely dominates the structure of the cubic for t  1. In the (large complex structure) limit t !1, the stere- 2 2 2 0 2 ographic cubic reduces to }3 ¼ }2ð}2 À p Þ¼ð}2Þ =4. 2 3 With ðX; Y Þ¼ð}2=p ;}3=p Þ, this gives a planar cubic

2 2 0 2 Y ¼ X ðX À 1Þ¼ðX Þ =4  CP2ðX; Y Þ ð18Þ Figure 10. Papillon plot of the stereographic planar cubic in that looks quite similar to (14). Equation (18) should be (9), with s ¼ it, t 2ð0:8; 2:5Þ, x 2ðÀ1; 2Þ, y 2ðÀ1; 1Þ (t compared to the Weierstrass cubic in this limit, increases out of the plane of the paper). ðX 0Þ2=4 ¼ X 3 À p4X=3 À 2p6=27, which is not particularly illuminating. which we can think of as the surface of a sphere, by The similarity between (14) and (18) is to some extent identifying all points at infinity. misleading, since we now are dealing with complex func- Figure 11(b) shows parametric plots of the smooth cubic tions, and we need to decompress the complex notation in 2 ð1 þ c1Þð1; c1Þ (purple) parameterized by c1 ¼ cot u1, and order to draw pictures of what is going on. 2 the nodal cubic ð1 À c2Þð1; c2Þ (orange) parameterized by With X ¼ x1 þ ix2 and Y ¼ y1 þ iy2, the two slicings c2 ¼ cot u2, flattened by drawing both in the same plane. discussed in the previous section become This is the t !1 member of the family whose portrait 2 3 2 (papillon plot) is shown in Figure 10. The point (0, 0) on x2 ¼ 0 ¼ y2)y1 ¼ x1 À x1 ; ð19Þ the nodal curve projects to c2 ¼Æ1 on the screen. The tail

2 2 3 of the node projects to points on the screen outside the x2 ¼ 0 ¼ y1)y ¼ x À x : ð20Þ 2 1 1 interval c2 2hÀ1; 1i. Unlike the projection of the smooth cubic, which is well defined everywhere, the nodal pro- We recognize (19) as the real smooth cubic that we studied jection is ill defined at the node and must be blown up at above in the section on stereography of circles, so this cycle this point. Since the mappings are bijective (except at must be the circle that has not degenerated. (0, 0)), this provides a birational map between the cubic Both cubics are plotted in Figure 11, which also shows nodal curve and the 2D screen. the stereographic projection of these cubics onto the 2D In summary, we have obtained a detailed and consistent plane x1 ¼ 1, which we can think of as a projection screen. picture of the geometry of the nodal cubic. The mero- 2 3 2 morphic 2D lattice functions labeling the point ð} ;} Þ on In Figure 11(a), the smooth cubic y1 ¼ x1 À x1 (purple) 2 3 2 2 3 a nodal planar cubic are somewhat peculiar complexified and the nodal cubic y2 ¼ x1 À x1 (orange) are orthogonal cycles that generate the nodal curve. To see this, we can stereographic coordinates. This allows us to conclude that think of all the endpoints as touching at infinity, repre- the nodal cubic is birational to a sphere. sented here by dashed circles intersecting in the point labeled 1. Sliding and shrinking the purple circle along the orange graph, to which it is attached at two points, gen- ACKNOWLEDGMENTS erates the nodal surface shown in the inset (a pinched The author is grateful for the hospitality of the CERN torus). Both cycles are projected onto the 2D plane x1 ¼ 1, Theoretical Physics Department (CERN-TH) near Geneva,

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Appendix: Modular Curve Geometry Up to normalization, G2ðsÞ is a holomorphic connection on the modular curve X. The covariant derivative with this connection generates the Ramanujan identities in (8). Let T ðnÀmÞ denote a mixed tensor with n contravariant (vector) indices and m covariant (covector) indices. Since these indices take only one value on a complex curve, they are usually suppressed. Under a modular coordinate transformation s ! s0, the holomorphic connection vðsÞ¼ 1 C11ðsÞ and k-tensors transform as vðsÞÀ!ðcs þ dÞ2vðsÞþ2cðcs þ dÞ ; T ðÆkÞ À! ð cs þ dÞÇ2k T ðÆkÞ ; and the covariant derivative with this connection is  o rT ðnÀmÞ ¼ þðn À mÞ v T ðnÀmÞ : os

Holomorphic modular k-tensors are called modular forms of weight w ¼Æ2k. The ring of all modular forms is gen- erated by E4 and E6. The renormalized Eisenstein function e2ðsÞ¼piE2ðsÞ=3 is a holomorphic modular connection, cf. (5), rather than a Figure 11. Stereography of a complex cubic. (a) The smooth holomorphic modular tensor (form). Furthermore, since cubic y2 ¼ x3 À x2 (purple) and the nodal cubic y2 ¼ x2 À x3 the difference between two connections is a tensor, by 1 1 1 2 1 1 s; s = s (orange) are orthogonal cycles that generate the nodal curve. subtracting the hyperbolic connection h2ð Þ¼i I on the upper half-plane, e can be traded for a quasiholo- (b) Parametric plots of the smooth cubic (purple) parameter- 2 morphic modular 2-form e^ ðs; sÞ¼e ðsÞÀh ðs; sÞ. ized by c ¼ cot u , and the nodal cubic (orange) 2 2 2 1 1 The only nonvanishing component of the holomorphic parameterized by c ¼ cot u , have been flattened by drawing 2 2 modular curvature 2-tensor (w ¼ 4) is x ¼ ov À v2=2 (with both in the same plane. o ¼ o=os), and since the space of forms of weight 4 is one- dimensional, xðv ¼ e2Þ must be proportional to E4. and the Quantum Computing Technologies Group (QCT) Matching constant terms gives at the Institute for High Energy Physics (IFAE) in Barcelona.  1 1 pi 2 x ¼ oe À e2 ¼À E : ð21Þ 2 2 2 2 3 4 FUNDING The covariant derivative of a k-tensor produces a ðk þ 1Þ- Open access funding provided by University of Oslo tensor, i.e., a form of weight w þ 2, from a form of weight (including Oslo University Hospital). w. The covariant derivative of x is therefore a 3-tensor, and

Ó 2021 The Author(s), Volume 43, Number 3, 2021 25 since the space of forms of weight 6 is one-dimensional, REFERENCES rx must be proportional to E6. Matching constant terms, [1] T. M. Apostol. Modular Functions and Dirichlet Series in Number we obtain Theory. Springer, 1990.  [2] M. Atiyah. The logarithm of the Dedekind g-function. Math. Ann. pi 3 rx ¼ ðÞo À 2e x ¼ E : ð22Þ 278 (1987), 335–380. 2 3 6 [3] R. Dedekind. Erla¨ uterungen zu zwei Fragmenten Riemann. In Similarly, the covariant derivative of rx produces a 4- Riemann’s Gesammelte Mathematische Werke 2, pp. 466–478. tensor, and since the space of forms of weight 8 is one- Dover, 1982. 2 2 dimensional, r x must be proportional to E4 . Matching [4] G. Eisenstein. Beitra¨ ge zur Theorie der elliptischen Functionen. constants gives Journal fu¨ r die Reine und Angewandte Mathematik 35 (1847),  153–184. pi 4 r2x ¼ ðÞo À 3e ðÞo À 2e x ¼À3 E2 : ð23Þ [5] G. H. Hardy. Divergent Series. Clarendon Press, 1949. 2 2 3 4 [6] E. Hecke. Theorie der Eisensteinschen Reihen ho¨ herer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik. Abh. When these geometric statements (tensor identities on the Math. Sem. Univ. Hamburg 5 (1927), 199–224. modular curve), (21)–(23), are rewritten in terms of E2 and [7] C. A. Lu¨ tken. Holomorphic anomaly in the quantum Hall system. À1 the derivation D ¼ o=o ln q ¼ð2piÞ o, we obtain the Nucl. Phys. B 759 [FS] (2006), 343–369. Ramanujan identities in (8). [8] C. A. Lu¨ tken. Elliptic mirror of the quantum Hall effect. Phys. Rev. B 99:19 (2019), 195152 (31). [9] S. Ramanujan. On certain arithmetical functions. Trans. Camb. C. A. Lu¨ tken Philos. Soc. 22:9 (1916), 159–184. Dept. of Physics [10] A. Weil. Elliptic Functions according to Eisenstein and Kronecker. University of Oslo Springer, 1976. Oslo Norway Quantum Research Center Publisher’s Note Springer Nature remains neutral with Technology Innovation Institute regard to jurisdictional claims in published maps and Abu Dhabi institutional affiliations. United Arab Emirates e-mail: [email protected]

26 THE MATHEMATICAL INTELLIGENCER