Elliptic Cohomology and Topological Modular Forms¶

Job: develop elliptic cohomology theories from elliptic curves (Landweber exactness), the Goerss–Hopkins–Miller sheaf $\mathcal{O}^{\mathrm{top}}$ on $\mathcal{M}_{\mathrm{ell}}$ and the $\mathrm{tmf}/\mathrm{TMF}/\mathrm{Tmf}$ variants, the Witten genus / string orientation, and the coefficient ring $\pi_*\mathrm{tmf}$ as integral modular forms.

Core¶

Elliptic cohomology theories¶

A complex-oriented elliptic cohomology theory $E^*$ is associated to an elliptic curve $C$ over a ring $R$: the formal group of $E$ is the formal completion $\hat C$ at the origin, with the formal group law $F_{\hat C}$ (the addition on $\hat C$). The coefficient ring $E^*=\mathcal{O}_C(\hat C\setminus\{0\})$ (functions on the punctured formal neighborhood), with periodicity from the elliptic logarithm $\log:\hat C\to\hat{\mathbb{G}}_a$.

Curve $C$	FGL	Coefficients	Periodicity
Tate curve $T_q$	multiplicative (formal $\hat{\mathbb{G}}_m$)	$\mathbb{Z}[[q]][u,u^{-1}]$	the $q$-expansion / Tate
Legendre $y^2=x(x-1)(x-\lambda)$	the Legendre FGL	$\mathbb{Z}[\lambda^{\pm}][u,u^{-1}]$	the $\lambda$-param
Weierstrass $y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$	the Weierstrass FGL	$\mathbb{Z}[a_1,a_2,a_3,a_4,a_6][u,u^{-1}]$	the universal Weierstrass
degenerate multiplicative ($\hat{\mathbb{G}}_m$)	multiplicative $x+y+xy$	$\mathbb{Z}[u^{\pm}]=K^*$	$K$-theory (the degenerate / nodal case)

The Landweber exact functor theorem¶

The Landweber exact functor theorem (LEFT): a formal group law $F$ over $R$ gives a cohomology theory $E^*(X)=\mathrm{MU}^*(X)\otimes_{\mathrm{MU}_*}R$ iff $R$ is Landweber exact over $\mathrm{MU}_*$ — i.e. for every prime $p$ and height $n$, multiplication by the $v_n$-generator $v_n^F$ is injective modulo $(p,v_1,\dots,v_{n-1})$. The Landweber–Ravenel–Stong construction: the FGL of $\hat C$ (the elliptic curve) is Landweber exact over $\mathbb{Z}[a_1,a_2,a_3,a_4,a_6]$ modulo the elliptic-curve relations (the Weierstrass FGL is Landweber exact), giving the elliptic cohomology theory.

The Ochanine / Witten theory: the elliptic genus (the Ochanine genus, a genus $\mathrm{MSO}\to R$ with values in modular forms), the Witten genus $W:\mathrm{MString}\to\mathbb{Z}[[q]]$ (the string orientation of the elliptic cohomology / the $\sigma$-orientation), and the rigidity theorems (the Witten genus of a string manifold with positive Ricci curvature vanishes, the Stolz–Teichner rigidity — the elliptic cohomology of a point with a string structure is a modular form).

Topological modular forms $\mathrm{tmf}$ and $\mathrm{TMF}$¶

The moduli stack of elliptic curves $\mathcal{M}_{\mathrm{ell}}$ (the Deligne–Mumford stack of smooth elliptic curves), and the Goerss–Hopkins–Miller sheaf $\mathcal{O}^{\mathrm{top}}$ (a sheaf of even-periodic $\mathbb{E}_\infty$-rings on the étale site of $\mathcal{M}_{\mathrm{ell}}$), with the stalk at $C$ the elliptic spectrum $E_C$.

Variant	Definition	Coefficients
$\mathrm{tmf}$ (connective)	$\mathrm{tmf}=\Gamma(\mathcal{M}_{\mathrm{ell}},\mathcal{O}^{\mathrm{top}})$ (the connective cover of the global sections)	$\pi_*\mathrm{tmf}$ = integral modular forms
$\mathrm{TMF}$ (periodic)	the periodic version, inverting the periodicity element $\Delta\in\pi_{24}\mathrm{tmf}$	$\pi_\mathrm{TMF}=\pi_\mathrm{tmf}[\Delta^{-1}]$
$\mathrm{Tmf}$ (compactified)	over $\overline{\mathcal{M}}_{\mathrm{ell}}$ (the moduli with the nodal / compactified elliptic curves)	the ring of modular forms with the cusp

The coefficient ring $\pi_*\mathrm{tmf}$ (Bauer, the Adams / Adams–Novikov computation): $\pi_*\mathrm{tmf}$ is the ring of integral modular forms (computed by the Adams–Novikov spectral sequence, the $E_2$-page the modular forms, the differentials the Hopf maps $\eta,\nu,\sigma$).

The descent / obstruction theory and the $\mathcal{O}^{\mathrm{top}}$ construction¶

The Goerss–Hopkins obstruction theory: the sheaf $\mathcal{O}^{\mathrm{top}}$ is constructed by the descent of $\mathbb{E}_\infty$-ring spectra on the étale site of $\mathcal{M}_{\mathrm{ell}}$, with the obstruction groups in the André–Quillen cohomology / the cotangent cohomology of the $\mathbb{E}_\infty$-rings. The Hopkins–Miller sheaf (the original construction via the Morava $E$-theory / the Landweber exact functor theorem, refined by Goerss–Hopkins to the $\mathbb{E}_\infty$-descent).

The arithmetic fracture: the gluing of $\mathrm{tmf}$ from the $p$-complete and the rational pieces, the pullback square $$ \mathrm{tmf}\to\mathrm{tmf}p\times) $$ (the arithmetic fracture square, the gluing of the }\otimes\mathbb{Q}}(\mathrm{tmf}\otimes\mathbb{Q$p$-adic and the rational / chromatic pieces).

See it¶

The Witten genus and $\pi_{2k}\mathrm{tmf}$. The Witten genus $W:\mathrm{MString}\to\mathbb{Z}[[q]]$ (the string-orientation of $\mathrm{tmf}$, the Ando–Hopkins–Rezk $\sigma$-orientation) maps the string cobordism to the modular forms: $$ W(M,\phi)=\big\langle\hat A(TM)\,\mathrm{ch}(\phi),[M]\big\rangle\in\mathbb{Z}[[q]], $$ with $\phi$ the string structure (the lift of the classifying map $M\to BSO$ to $B\mathrm{String}$, the 2-connected cover). The rigidity theorem (the Stolz–Teichner / the Witten rigidity): the Witten genus of a string manifold with positive Ricci curvature vanishes (the elliptic cohomology is rigid on positive-curvature string manifolds).

The $\pi_{2k}\mathrm{tmf}$ are the integral modular forms (with the $q$-expansion): - $\pi_0\mathrm{tmf}=\mathbb{Z}$ (the modular forms of weight 0, the constants / the $j$-invariant ring); - $\pi_{24}\mathrm{tmf}$ contains the discriminant $\Delta(q)=q\prod_{n\ge1}(1-q^n)^{24}$ (the modular form of weight 12, the $q$-expansion), the periodicity element of $\mathrm{TMF}$; - $\pi_{8}\mathrm{tmf}$ contains the Eisenstein series $E_4(q)=1+240\sum_{n\ge1}\sigma_3(n)q^n$ (the modular form of weight 4, the $E_4$); - $\pi_{16}\mathrm{tmf}$ contains the Eisenstein series $E_6$ (the modular form of weight 6, up to the torsion relations).

The key point: the topological modular forms recover the integral modular forms as the homotopy groups, and the Witten genus is the string orientation — the deep connection between stable homotopy, elliptic curves, and modular forms. The $\Delta$ function (the discriminant) is the periodicity of $\mathrm{TMF}$, the $E_4$ is the $\pi_8$ class, and the partition function of the heterotic string (the $q$-expansion of the Witten genus) is the prototypical $\mathrm{tmf}$ computation.

Check¶

Why does the string structure (the 2-connected cover $B\mathrm{String}\to BSO$) appear in the orientation of $\mathrm{tmf}$, and what fails for spin?

The orientation of an even-periodic $\mathbb{E}_\infty$-ring $E$ (the Thom class / the $\sigma$-orientation, Ando–Hopkins–Rezk) requires the classifying map $M\to BO$ to lift to $B\mathrm{String}$ (the 2-connected cover, killing $\pi_1, \pi_2$ of $BO$, i.e. the $w_1, w_2$ and the $p_1/2$). The obstruction is the half-Pontryagin class $p_1/2\in H^4(M,\mathbb{Z})$ (the String class); for a spin manifold ($w_1=w_2=0$) the $p_1/2$ need not vanish, so the spin orientation of $\mathrm{tmf}$ fails. The string structure ($p_1/2=0$) is exactly the condition for the $\mathrm{tmf}$ orientation / the Witten genus, the lift to $B\mathrm{String}$.

Verify that $\pi_0\mathrm{tmf}=\mathbb{Z}$ and $\pi_{24}\mathrm{tmf}\supset\mathbb{Z}\{\Delta\}$ are consistent with the integral modular forms.

$\pi_0\mathrm{tmf}$ is the global sections of $\mathcal{O}^{\mathrm{top}}$ in degree 0, the integral modular forms of weight 0 (the constants $\mathbb{Z}$, since the only weight-0 modular forms are the constants, the $j$-invariant is a rational function not a form). $\pi_{24}\mathrm{tmf}$ is the modular forms of weight 12 (the degree-24 = weight-12 by the periodicity convention), and $\Delta$ (the discriminant, weight 12) generates a $\mathbb{Z}$-summand in $\pi_{24}\mathrm{tmf}$ (with the torsion relations from the Hopf maps). The integrality (the $\mathbb{Z}$-summand, not the $\mathbb{Q}$-vector space) is the content of the Bauer computation: $\pi_*\mathrm{tmf}\otimes\mathbb{Q}=\mathbb{Q}[E_4,E_6]$ (the rational modular forms), and the integral $\pi_*\mathrm{tmf}$ is the integral modular forms (a subring of $\mathbb{Z}[E_4,E_6,\Delta]$ with the relation $E_4^3-E_6^2=1728\Delta$).

Pitfalls¶

The Landweber exact functor theorem is the construction route for elliptic cohomology (the FGL exactness), but not every elliptic curve is Landweber exact — the non-Landweber cases (the supersingular curves, where the FGL has height $>1$ and the $v_n$ injectivity fails) need the Goerss–Hopkins–Miller descent (the $\mathcal{O}^{\mathrm{top}}$ sheaf via the obstruction theory, not the LEFT).
The $\mathcal{O}^{\mathrm{top}}$ sheaf / $\mathrm{tmf}$ is a deep construction (the Goerss–Hopkins obstruction theory, the André–Quillen / cotangent cohomology obstructions), not a formal sheafification of the elliptic spectra.
$\pi_*\mathrm{tmf}$ is computed (Bauer) but the full $\mathrm{tmf}$ has torsion (the Hopf maps $\eta,\nu,\sigma$ kill classes, the $\pi_*\mathrm{tmf}$ has 2- and 3-torsion), and the $\mathrm{TMF}$ is the periodic version (inverting $\Delta$) — do not conflate $\mathrm{tmf}$ (connective), $\mathrm{TMF}$ (periodic), $\mathrm{Tmf}$ (compactified).
The Witten genus is the string orientation (the orientation requires the string structure, not just a spin / orientation); the spin version (the Ochanine genus) is the MSO genus, not the string / tmf orientation.
The arithmetic fracture glues $\mathrm{tmf}$ from the $p$-complete and rational pieces, but the gluing is non-trivial (the $\mathrm{tmf}\to\mathrm{tmf}_p\times_{\mathrm{tmf}_\mathbb{Q}}\mathrm{tmf}\otimes\mathbb{Q}$ square is a pullback only after the appropriate completion, the arithmetic square).
The rigidity theorem (the Witten genus of a positive-Ricci-curvature string manifold vanishes) is a theorem (the Stolz–Teichner rigidity), not a formal consequence of the orientation.

Refs¶

Landweber, "Elliptic cohomology" (1988); Landweber–Ravenel–Stong, "Periodic cohomology theories" (1993)
Goerss–Hopkins, "Moduli spaces of commutative ring spectra" (2005); Hopkins–Miller (the $\mathcal{O}^{\mathrm{top}}$ sheaf, unpublished lectures)
Ando–Hopkins–Rezk, "Multiplicative orientations of KO and tmf" (2010); Behrens, "The construction of tmf" (2011)
Bauer, "Computation of the homotopy of the spectrum tmf" (2003); Rezk, "Notes on the Hopkins–Miller theorem"
Witten, "Elliptic genera and quantum field theory" (1988); Stolz–Teichner, "The elliptic Witten genus"

Next¶

SAG index · Foundations · Derived Moduli & Deformation · Spectral GAGA & Descent · Chromatic & Morava E-theory · Motivic / algebraic cobordism · Condensed mathematics

Curve \(C\)	FGL	Coefficients	Periodicity
Tate curve \(T_q\)	multiplicative (formal \(\hat{\mathbb{G}}_m\))	\(\mathbb{Z}[[q]][u,u^{-1}]\)	the \(q\)-expansion / Tate
Legendre \(y^2=x(x-1)(x-\lambda)\)	the Legendre FGL	\(\mathbb{Z}[\lambda^{\pm}][u,u^{-1}]\)	the \(\lambda\)-param
Weierstrass \(y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6\)	the Weierstrass FGL	\(\mathbb{Z}[a_1,a_2,a_3,a_4,a_6][u,u^{-1}]\)	the universal Weierstrass
degenerate multiplicative (\(\hat{\mathbb{G}}_m\))	multiplicative \(x+y+xy\)	\(\mathbb{Z}[u^{\pm}]=K^*\)	\(K\)-theory (the degenerate / nodal case)

Variant	Definition	Coefficients
\(\mathrm{tmf}\) (connective)	\(\mathrm{tmf}=\Gamma(\mathcal{M}_{\mathrm{ell}},\mathcal{O}^{\mathrm{top}})\) (the connective cover of the global sections)	\(\pi_*\mathrm{tmf}\) = integral modular forms
\(\mathrm{TMF}\) (periodic)	the periodic version, inverting the periodicity element \(\Delta\in\pi_{24}\mathrm{tmf}\)	\(\pi_\mathrm{TMF}=\pi_\mathrm{tmf}[\Delta^{-1}]\)
\(\mathrm{Tmf}\) (compactified)	over \(\overline{\mathcal{M}}_{\mathrm{ell}}\) (the moduli with the nodal / compactified elliptic curves)	the ring of modular forms with the cusp