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 ½}; 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=dz. 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).