03.06.26 · modern-geometry / characteristic-classes

Pointer: elliptic cohomology

shipped3 tiersLean: none

Anchor (Master): Quillen 1969 *Bull. AMS* 75, 1293-1298; Landweber 1976 *Amer. J. Math.* 98, 591-610; Landweber-Ravenel-Stong 1995 *AMS Cont. Math.* 181, 317-337; Goerss-Hopkins 2004 *Lond. Math. Soc. LNS* 315, 151-200; Ando-Hopkins-Strickland 2001 *Invent. Math.* 146, 595-687; Hopkins 2002 ICM Vol. I, 291-317; Lurie 2009 *Curr. Devel. Math.* 2008, 219-277; Stolz-Teichner 2011 AMS PSPM 83, 279-340; Behrens et al. eds. 2014 *Topological Modular Forms*, AMS Math. Surveys & Monographs 201

Intuition Beginner

Ordinary cohomology assigns to each space a sequence of abelian groups, and the recipe runs in a way that respects an addition rule: the cohomology of a circle plus the cohomology of another circle splices together by simple addition. Topological $K$ -theory does the same but with multiplication in place of addition; the tensor product of complex line bundles obeys a multiplicative law. The question elliptic cohomology answers is: can the same kind of cohomology theory be built whose internal arithmetic comes from an elliptic curve instead?

The answer is yes. Each elliptic curve over a ring carries a small piece of formal arithmetic attached to its identity element, and this arithmetic plays the role that addition plays for ordinary cohomology and multiplication plays for $K$ -theory. The resulting theory assigns to each space a graded module over a ring of modular forms, and the modular-form-valued invariants of a manifold include the Witten genus discussed in the modularity unit.

The reason this matters is that the elliptic theory unifies geometric data that looked unrelated. The signature, the Dirac index, and the Witten genus all sit inside one structured object, and the way they fit together becomes a piece of arithmetic geometry rather than a list of separate calculations.

Visual Beginner

A horizontal tower drawn schematically: at the leftmost rung sits ordinary cohomology, labelled height $0$ . One step to the right sits complex $K$ -theory, labelled height $1$ . The next rung is elliptic cohomology, labelled height $2$ . Further to the right sits a series of higher-height theories. Above the tower, a single curving arrow connects every rung to a common origin labelled "formal group law of complex cobordism" — the universal recipe from which each theory pulls its piece of arithmetic.

The picture captures the central idea: a hierarchy of cohomology theories, organised by an integer called the chromatic height, each attached to a different kind of formal arithmetic. Elliptic cohomology occupies the height- $2$ slot, and the modular-form-valued invariants of a string manifold are the natural arithmetic at that height.

Worked example Beginner

Pick a familiar four-real-dimensional shape: a K3 surface, the unique simply connected closed shape of real dimension $4$ that is spin and has vanishing first integer Pontryagin class. Its Witten genus is a specific modular form for the full modular group, and its elliptic-cohomology refinement gives a specific element of the topological-modular-form ring in degree $4$ .

Step 1. The Witten genus of a K3 surface, expanded as a power series in $q$ , starts $2 - 48 q + \dots$ . The leading coefficient $2$ is the Dirac index of the K3 surface, the same answer one gets by counting harmonic spinors with sign.

Step 2. The whole power series is the Fourier expansion of a specific modular form for the full modular group: namely $2 E_{4} (τ) \cdot E_{6} (τ) /Δ (τ)$ in standard notation, restricted to its piece of weight $2$ , with $E_{4}, E_{6}$ the Eisenstein series and $Δ$ the discriminant.

Step 3. The topological-modular-form ring assigns to the K3 surface a class in its degree- $4$ piece. The class refines the modular-form-valued Witten genus to a topological invariant.

Step 4. Comparing the topological-modular-form ring with the ring of integer modular forms after multiplication by $1/6$ , the two rings agree, so the K3 invariant maps to the integer modular form attached to the K3 surface.

What this tells us: a four-dimensional shape produces a single invariant living in three places at once — a Witten genus, an integer modular form, and a class in the topological-modular-form ring — and the comparison map between them is an isomorphism after a small denominator.

Check your understanding Beginner

Formal definition Intermediate+

A (generalised) cohomology theory $E^{*}$ is a sequence of contravariant functors from pointed spaces to abelian groups satisfying the Eilenberg-Steenrod axioms minus the dimension axiom; equivalently it is represented by a spectrum $E$ in the stable homotopy category (Brown representability). A theory $E^{*}$ is complex-orientable if there exists a class $x \in E^{2} (CP^{\infty})$ restricting to a generator of $E^{2} (S^{2}) = E^{0} (pt)$ ; the coefficient ring is denoted $E^{*} = E^{*} (pt)$ .

Definition (formal group law). A one-dimensional commutative formal group law over a commutative ring $R$ is a formal power series $F (X, Y) \in R [[X, Y]]$ satisfying (i) $F (X, 0) = X$ and $F (0, Y) = Y$ , (ii) $F (X, Y) = F (Y, X)$ , (iii) $F (X, F (Y, Z)) = F (F (X, Y), Z)$ . The two simplest examples are the additive law $F_{a} (X, Y) = X + Y$ and the multiplicative law $F_{m} (X, Y) = X + Y + X Y$ . For a one-dimensional commutative algebraic group $G$ over $R$ with origin $e$ and a chosen local coordinate $X$ at $e$ , the multiplication $G \times G \to G$ expanded in the local coordinate produces a formal group law over $R$ .

Definition (Lazard ring). The Lazard ring $L$ is the universal ring for one-dimensional commutative formal group laws: it is the quotient of the polynomial ring $Z [a_{ij} : i, j \geq 1]$ by the ideal generated by the relations forcing the abstract series $F_{univ} (X, Y) = X + Y + \sum_{i, j \geq 1} a_{ij} X^{i} Y^{j}$ to satisfy (i)-(iii). Lazard 1955 proved $L ≅ Z [x_{1}, x_{2}, \dots]$ as an abstract ring, with $x_{i}$ in degree $2 i$ when one grades by $de g (a_{ij}) = 2 (i + j - 1)$ .

Definition (formal group law of a complex-orientable theory). Given a complex-orientable theory $E$ with orientation $x$ , the line-bundle map $CP^{\infty} \times CP^{\infty} \to CP^{\infty}$ classifying the tensor product of line bundles pulls $x$ back to a class $F_{E} (x_{1}, x_{2}) \in E^{*} (CP^{\infty} \times CP^{\infty}) = E^{*} [[x_{1}, x_{2}]]$ , where $x_{i}$ is the pullback of $x$ along the $i$ -th projection. The series $F_{E} (X, Y)$ is the formal group law of $E$ over $E^{*}$ . For ordinary integer cohomology $F_{E} = F_{a}$ is additive; for complex $K$ -theory $F_{E} = F_{m}$ is multiplicative.

Definition (elliptic spectrum, after Landweber-Ravenel-Stong). An elliptic spectrum is a triple $(E, C, t)$ where $E$ is an even-periodic homotopy commutative ring spectrum, $C$ is a generalised elliptic curve over $E_{0} = E^{0} (pt)$ , and $t : C \sim Spf (E^{0} (CP^{\infty}))$ is an isomorphism between the formal group of $C$ at the origin and the formal group of $E$ . The motivating example is the periodic complex cohomology theory attached to a Weierstrass elliptic curve over $Z [1/6]$ via the Landweber exact-functor theorem.

Definition ( $TMF$ ). The topological-modular-form spectrum $TMF$ is the global sections of a sheaf $O^{top}$ of $E_{\infty}$ -ring spectra on the moduli stack $M_{ell}$ of generalised elliptic curves, constructed by Hopkins-Miller and refined by Lurie; the underlying homotopy sheaf encodes the structure that on each étale chart $Spec (R) \to M_{ell}$ classifying an elliptic curve $C / R$ , the section $O^{top} (R)$ is an elliptic spectrum with formal group law the formal group of $C$ .

Key theorem with proof Intermediate+

Theorem (Quillen 1969). The natural ring map $\theta : L \to MU^(\mathrm{pt}) $c l a ss i f y in g t h e f or ma l g r o u pl a w o f co m pl e x co b or d i s m$ MU$ over the Lazard ring is an isomorphism of graded rings.* (Quillen 1969 Bull. AMS 75, 1293-1298 ^{[source pending]}.)

Proof (sketch). The proof has two halves.

Surjectivity. Milnor computed $M U^{*} (pt) = Z [m_{1}, m_{2}, \dots]$ as a polynomial ring on generators $m_{i}$ in degree $2 i$ , with $m_{i}$ a polynomial in the Chern numbers of $CP^{i}$ . The formal group law of $M U$ assembles to a series whose coefficients $a_{ij}$ are polynomial expressions in the $m_{k}$ . A direct degree count shows that the map $θ$ sends the Lazard generators in degree $2 i$ to elements whose image generates the rank- $1$ piece of $M U^{2 i} (pt)$ in that degree; surjectivity follows by induction on degree.

Injectivity. The reverse direction uses the fact that the formal group law of $M U$ is universal — every complex-orientable theory pulls its formal group law from the $M U$ orientation. If a Lazard relation held in $M U^{*} (pt)$ but not in $L$ , then the universal property of $L$ would be contradicted by the existence of a complex-orientable theory realising the unrelated relation. The argument is reduced to Lazard's 1955 classification of formal group laws, and the Lazard ring's universal property combined with the Milnor computation pins the map $θ$ as an isomorphism.

Combined statement. Both halves match degree by degree, so $θ$ is a graded isomorphism. $□$

Theorem (Landweber 1976 exact-functor theorem). Let $L \to R$ be a ring map. If for every prime $p$ the sequence $(p, v_{1}, v_{2}, \dots)$ of Hazewinkel generators of the ideal in $L_{(p)}$ acts regularly on $R$ , then the functor $X \mapsto MU_(X) \otimes_L R$ is a homology theory.* (Landweber 1976 Amer. J. Math. 98, 591-610 ^{[source pending]}.)

Proof (idea). The functor $X \mapsto M U_{*} (X) \otimes_{L} R$ is a half-exact functor automatically; the question is whether it satisfies the long exact sequence axiom. This reduces to checking that $Tor_{*}^{L} (M U_{*} (X), R)$ vanishes for $* \geq 1$ and every $X$ . The vanishing is verified after $p$ -localisation for each prime $p$ separately, where the structure of $M U_{*}$ as a $B P_{*}$ -comodule converts the question into the regular-sequence criterion on the Hazewinkel generators. Once $Tor_{*}^{L} (M U_{*} (X), R) = 0$ holds for every $X$ , exactness of the original functor follows from the exactness of $M U_{*} (-)$ together with the flatness consequence of regularity. $□$

Theorem (existence of elliptic cohomology; Landweber-Ravenel-Stong 1995). Let $C$ be a Weierstrass elliptic curve over a ring $R$ , with formal group law $F_{C}$ . If the ring map $L \to R$ classifying $F_{C}$ satisfies the Landweber regularity criterion, then there is an even-periodic homology theory $E\ell\ell^C_(X) = MU_*(X) \otimes_L R $w h ose f or ma l g r o u pl a w o v er$ R $i s$ F_C$.* (Landweber-Ravenel-Stong 1995 AMS Cont. Math. 181, 317-337 ^{[source pending]}.)

Proof (application). Apply the Landweber exact-functor theorem to the ring map $L \to R$ classifying the formal group law of $C$ . The Landweber regularity criterion is verified for Weierstrass elliptic curves over $Z [1/6]$ by direct computation (Landweber-Ravenel-Stong 1995): the Hazewinkel generators $v_{n}$ act regularly on the universal Weierstrass ring with $1/6$ inverted, since the discriminant $Δ$ becomes a unit after inverting $6$ and the height of the formal group of an elliptic curve at any prime $p > 3$ is $\leq 2$ , automatically forcing $v_{n}$ regularity for $n \geq 3$ . The corresponding even-periodic homology theory is the announced elliptic-cohomology theory attached to $C$ . $□$

Bridge. The Quillen theorem builds toward the comparison theorem that identifies $TMF^{*} (pt) \otimes Z [1/6]$ with integer modular forms, and appears again in the synthesis section as the foundational reason that complex cobordism organises every formal-group-theoretic cohomology theory. The foundational reason elliptic cohomology exists is exactly that the formal group law of an elliptic curve, after inverting $6$ , satisfies the Landweber regularity condition, and Landweber's theorem then converts the formal-group data into a cohomology theory. This is exactly the same mechanism that produces complex $K$ -theory from the multiplicative formal group law and ordinary cohomology from the additive one — the bridge is that all three are base-changes of complex cobordism along ring maps $L \to R$ , and the formal group law over $R$ is the only piece of data that the resulting theory remembers about the chosen orientation. Putting these together, the central insight is that complex cobordism is the universal complex-orientable cohomology theory, and every other complex-orientable theory generalises this universal one along a chosen formal group law. The chromatic-height filtration on the universal formal group law identifies a hierarchy of theories with heights $0, 1, 2, \dots$ : ordinary cohomology, $K$ -theory, elliptic cohomology, Morava $E$ -theories. The same pattern recurs in the construction of $TMF$ , where the moduli stack of elliptic curves replaces a single elliptic curve and the sheaf of $E_{\infty}$ -rings assembles the local elliptic spectra into a single global object.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

The Weierstrass elliptic curve $y^{2} = 4 x^{3} - g_{2} x - g_{3}$ over a ring $R$ has formal group law in the local coordinate $z = - 2 y / x$ at the origin. Write down the first three terms of the formal group law.

Hint

Use the addition formula on the elliptic curve to compute $z_{1} \oplus z_{2}$ as a power series in $z_{1}, z_{2}$ with coefficients in $R$ . The leading terms are $z_{1} + z_{2}$ and the cubic correction involves $g_{2}$ .

Answer

The formal group law has the form $$ F_C(z_1, z_2) = z_1 + z_2 - \frac{g_2}{20}\big( z_1 z_2^3 + z_1^3 z_2 \big) - \frac{g_3}{14}\big( z_1 z_2^4 + z_1^4 z_2 \big) + \cdots, $$ where the lower-order corrections vanish and the leading correction is degree $4$ . The exact coefficients can be derived from the Weierstrass addition formula, and the appearance of $g_{2}$ and $g_{3}$ as denominators is the reason the Landweber criterion is verified after inverting $6$ — the denominators $20 = 4 \cdot 5$ and $14 = 2 \cdot 7$ both become units once one inverts the discriminant $Δ = g_{2}^{3} - 27 g_{3}^{2}$ and the small primes. Rubric: full credit for the first non-additive correction at degree $4$ with the $g_{2} /20$ coefficient.

Exercise 4 (medium, short-answer).

State the Landweber regularity criterion for a ring map $θ : L \to R$ , and explain why it fails for $L \to F_{p}$ given by the additive formal group law.

Hint

The Hazewinkel generators $v_{n} \in B P_{*}$ pull back from a regular sequence on the Brown-Peterson part of $L$ . The additive law has height $\infty$ , equivalently $v_{n} = 0$ for every $n$ after $p$ -localisation.

Answer

The Landweber regularity criterion is: for every prime $p$ , the sequence $(p, v_{1}, v_{2}, v_{3}, \dots)$ acts regularly on $R$ , where $v_{n} \in L$ is a chosen lift of the Hazewinkel generator. Regular means: $p$ is a non-zero-divisor in $R$ ; $v_{1}$ is a non-zero-divisor in $R / p$ ; $v_{2}$ is a non-zero-divisor in $R / (p, v_{1})$ ; and so on.

For the additive formal group law over $F_{p}$ , the formal group law $F_{a} (X, Y) = X + Y$ has all $v_{n}$ equal to $0$ in $F_{p}$ (the $p$ -series of $F_{a}$ is $[p] (X) = pX = 0$ in $F_{p}$ , so the height is $\infty$ ). The first step of the regular-sequence check requires $p$ to be a non-zero-divisor in $F_{p}$ , but $p$ is already zero, so the criterion fails immediately. Rubric: full credit for the regular-sequence statement and the observation that $p = 0$ in $F_{p}$ .

Exercise 5 (medium, short-answer).

Sketch why complex $K$ -theory is the elliptic cohomology theory attached to the multiplicative formal group law (rather than the formal group law of an elliptic curve).

Hint

Complex $K$ -theory has formal group law $F (X, Y) = X + Y + X Y$ , which is the formal group of the multiplicative algebraic group $G_{m}$ . The elliptic theory at height $1$ specialises to $K$ -theory at the cusp where the elliptic curve degenerates to a nodal cubic.

Answer

The multiplicative formal group law $F_{m} (X, Y) = X + Y + X Y$ is the formal group of the algebraic group $G_{m}$ in the local coordinate $X = T - 1$ near $T = 1$ . The Weierstrass family of elliptic curves has a degeneration locus along the discriminant $Δ = 0$ , and at the nodal cubic (a special fibre of this degeneration) the formal group is exactly $G_{m}$ . So complex $K$ -theory is the cohomology theory at the boundary of the moduli stack of elliptic curves where the curve degenerates to a nodal cubic. The chromatic-height picture is consistent: $K$ -theory has height $1$ , ordinary elliptic curves have formal-group height $1$ , and the cusp of the moduli stack is where $K$ -theory sits. Rubric: full credit for the nodal-cubic / $G_{m}$ identification.

Exercise 6 (medium, short-answer).

Why does the comparison map $tmf_{*} \to MF_{*}$ to integer modular forms become an isomorphism only after inverting $6$ ?

Hint

At the primes $2$ and $3$ , $tmf_{*}$ has torsion that does not lift to a modular-form refinement. The Weierstrass model of an elliptic curve degenerates at $2$ and $3$ in ways that produce non-modular-form classes in $tmf_{*}$ .

Answer

The comparison map is constructed via the global sections of the sheaf $O^{top}$ on the moduli stack $M_{ell}$ . Away from the primes $2$ and $3$ , the stack is well-behaved and the global-sections computation reproduces the classical ring of integer modular forms via the descent spectral sequence collapsing in low filtration. At $p = 2$ and $p = 3$ , the stack has substantial $2$ - and $3$ -torsion in its coherent cohomology, and the descent spectral sequence acquires non-zero differentials and extensions. These produce torsion classes in $tmf_{*}$ that have no modular-form counterpart. Inverting $6$ kills both the $2$ -torsion and the $3$ -torsion, and the comparison map becomes an isomorphism (Hopkins 2002 ICM Vol. I, 291-317). Rubric: full credit for the moduli-stack / coherent-cohomology mention with the $2$ - and $3$ -torsion explanation.

Exercise 7 (hard, short-answer).

The $σ$ -orientation $σ : MString \to TMF$ of Ando-Hopkins-Strickland realises the Witten genus as a map of ring spectra. State precisely what extra structure on the formal group of an elliptic curve is required for the orientation to exist, and why a string structure on a manifold is the topological counterpart.

Hint

The required structure is a cubical structure on the formal group law of the elliptic curve, in the sense of the theorem of the cube. Topologically, this corresponds to the vanishing of obstructions in the Whitehead tower, namely $w_{1}, w_{2}, p_{1} /2$ .

Answer

Ando-Hopkins-Strickland 2001 (Invent. Math. 146, 595-687) ^{[source pending]} prove that an $E_{\infty}$ -orientation $M U ⟨ 6 ⟩ \to E$ from a Thom spectrum $M U ⟨ 6 ⟩$ (with structure group reducing through the $5$ -connected cover of $U$ ) into an even-periodic ring spectrum $E$ with formal group $C$ is equivalent data to a cubical structure on the formal group: a trivialisation of the line bundle $Θ^{3} (L)$ on $C^{\times 3}$ , where $L$ is the rigidified line bundle on $C$ associated to the formal group law and $Θ^{3}$ is the cubical-difference functor of Mumford. The theorem of the cube ensures the existence of cubical structures on the universal elliptic curve over the moduli stack; the orientation lifts these existence statements to ring-spectrum maps.

Topologically, the $5$ -connected cover of $U$ is the string group $String$ , characterised by $π_{0} = π_{1} = π_{2} = π_{3} = 0$ . A string structure on a manifold $M$ is a lift of the tangent classifying map through the string-group fibration, equivalently the vanishing of $w_{1} (M), w_{2} (M)$ , and $p_{1} (M) /2$ . The first two are the orientation and spin conditions; the third is the obstruction to lifting from $Spin$ to $String$ . The cubical structure on the formal group of the elliptic curve is the algebraic shadow of this string condition: both record the trivialisation of a degree- $3$ obstruction class. Rubric: full credit for naming cubical structures, citing the theorem of the cube, and identifying string structure with the $p_{1} /2 = 0$ Whitehead-tower lift.

Exercise 8 (hard, short-answer).

Outline how Lurie's derived-algebraic-geometry construction of $TMF$ (Lurie 2009 Curr. Devel. Math. 2008, 219-277 ^{[source pending]}) differs from the Hopkins-Miller obstruction-theoretic construction, and why both produce the same ring spectrum.

Hint

The Hopkins-Miller construction uses Goerss-Hopkins obstruction theory to refine a homotopy ring spectrum to an $E_{\infty}$ -ring spectrum on each étale chart of $M_{ell}$ and then sheafifies. Lurie defines a derived stack $M_{ell}^{der}$ of oriented derived elliptic curves, proves its representability as a derived Deligne-Mumford stack, and takes global sections directly.

Answer

The Hopkins-Miller construction (Hopkins-Miller 1994-1999 ^{[source pending]}; Behrens et al. eds. 2014 ^{[source pending]}) is bottom-up. For each étale chart $Spec (R) \to M_{ell}$ classifying an elliptic curve $C / R$ , the Landweber-Ravenel-Stong construction produces a homotopy ring spectrum. Goerss-Hopkins obstruction theory (Goerss-Hopkins 2004 ^{[source pending]}) refines this to an $E_{\infty}$ -ring spectrum, with obstructions in André-Quillen cohomology that vanish for elliptic spectra by direct computation. The collection of $E_{\infty}$ -rings on étale charts assembles to a sheaf $O^{top}$ of $E_{\infty}$ -rings on $M_{ell}$ , and $TMF$ is the global sections.

The Lurie construction (Lurie 2009 ^{[source pending]}) is top-down. Lurie defines a moduli problem in derived algebraic geometry: the functor that assigns to a connective $E_{\infty}$ -ring $A$ the space of oriented derived elliptic curves over $A$ , an elliptic curve in the derived sense together with a compatibility between its formal group and the formal group of $A$ . He proves that this functor is representable by a derived Deligne-Mumford stack $M_{ell}^{der}$ whose underlying classical stack is $M_{ell}$ . The spectrum $TMF$ is the global sections of the structure sheaf of $M_{ell}^{der}$ .

Both constructions produce the same ring spectrum because the Hopkins-Miller sheaf $O^{top}$ is canonically isomorphic to the structure sheaf of $M_{ell}^{der}$ — Lurie's derived moduli problem is the universal home for the local elliptic spectra that Hopkins-Miller constructs on each chart. The Lurie construction has the conceptual advantage of replacing the obstruction-theoretic computation with a moduli-theoretic existence theorem; the Hopkins-Miller construction has the computational advantage that the obstruction groups are explicit and the resulting computation of $tmf_{*}$ proceeds along the descent spectral sequence with concrete differentials. Rubric: full credit for naming both constructions, identifying the conceptual difference (obstruction theory versus representable derived moduli), and explaining the equivalence via the universal property.

Advanced results Master

Theorem (Quillen's theorem; Quillen 1969). The natural ring map $L \to MU^(\mathrm{pt})$ classifying the formal group law of complex cobordism is an isomorphism of graded rings (Quillen 1969 Bull. AMS 75, 1293-1298 ^{[source pending]}).*

This is the foundational identification on which the entire complex-orientable / chromatic / elliptic picture rests. Every complex-orientable cohomology theory $E$ admits a ring map $M U \to E$ realising its complex orientation, equivalently a ring map $L = M U^{*} (pt) \to E^{*} (pt)$ classifying the formal group law of $E$ . The classification of cohomology theories with a complex orientation reduces to the algebraic classification of formal group laws, which is governed by the Lazard ring and (after $p$ -localisation) by the Brown-Peterson spectrum $B P$ with $B P_{*} = Z_{(p)} [v_{1}, v_{2}, \dots]$ .

Theorem (Landweber exact-functor theorem; Landweber 1976). Let $θ : L \to R$ be a ring map. If the sequence $(p, v_{1}, v_{2}, \dots)$ of Hazewinkel generators (lifts of the corresponding $BP_ $g e n er a t or s) a c t sr e g u l a r l y o n$ R_{(p)} $f or e v er y p r im e$ p $, t h e n$ X \mapsto MU_*(X) \otimes_L R$ defines a homology theory on the category of spectra (Landweber 1976 Amer. J. Math. 98, 591-610 ^{[source pending]}).*

The criterion is the algebraic counterpart of half-exactness: it ensures the higher $Tor$ groups vanish, hence the tensor product is exact in the appropriate sense, hence the resulting functor satisfies the long exact sequence axiom. The criterion is verified for Weierstrass elliptic curves over $Z [1/6]$ (Landweber-Ravenel-Stong 1995 AMS Cont. Math. 181, 317-337 ^{[source pending]}), and the resulting periodic cohomology theory is the elliptic cohomology of the curve.

Theorem (Hopkins-Miller). There exists a sheaf $O^{top}$ of $E_{\infty}$ -ring spectra on the moduli stack $M_{ell}$ of generalised elliptic curves over $Z$ whose underlying homotopy sheaf is the sheaf of elliptic spectra, and the global-sections spectrum $TMF = R Γ (M_{ell}, O^{top})$ is the topological-modular-form spectrum (Hopkins-Miller 1994-1999 ^{[source pending]}; Behrens et al. eds. 2014 ^{[source pending]}).

The construction proceeds in two steps. First, on each étale chart $Spec (R) \to M_{ell}$ classifying an elliptic curve $C / R$ , Landweber exactness produces a homotopy ring spectrum $E_{R}$ with formal group law the formal group of $C$ . Second, Goerss-Hopkins obstruction theory (Goerss-Hopkins 2004 Lond. Math. Soc. LNS 315, 151-200 ^{[source pending]}) refines $E_{R}$ to an $E_{\infty}$ -ring spectrum: the moduli space of $E_{\infty}$ -structures on a given homotopy ring spectrum has obstructions in André-Quillen cohomology of the homotopy ring, and these obstructions vanish for elliptic spectra by explicit computation. The étale-local $E_{\infty}$ -rings glue to a sheaf because the obstruction-theory gluing data are also obstruction-controlled and vanish for the same reasons.

Theorem (comparison with integer modular forms; Hopkins 2002). There is a natural ring map $\mathrm{tmf}_ \to \mathrm{MF}_ $f r o m t h e h o m o t o p y g r o u p so f t h eco nn ec t i v e t o p o l o g i c a l - m o d u l a r - f or m s p ec t r u m t o t h e g r a d e d r in g o f in t e g er m o d u l a r f or m s; t h e ma p b eco m es ani so m or p hi s ma f t er in v er t in g$ 6$ (Hopkins 2002 ICM Vol. I, 291-317 ^{[source pending]}).

The map is constructed via the descent spectral sequence for the sheaf $O^{top}$ on $M_{ell}$ . The $E_{2}$ -page is the cohomology of the moduli stack with coefficients in the line bundle of weight- $k$ modular forms, and the global-sections term in degree zero is the ring of integer modular forms. At primes $\geq 5$ the spectral sequence collapses; at $p = 2$ and $p = 3$ it has substantial higher differentials and extensions, which contribute torsion classes to $tmf_{*}$ that do not exist on the modular-form side.

Theorem (Ando-Hopkins-Strickland 2001 $σ$ -orientation). There is a ring-spectrum map $σ : MString \to TMF$ realising the Witten genus on string manifolds; equivalently, an $E_{\infty}$ -orientation of the topological-modular-form spectrum compatible with the formal group of the universal elliptic curve, classified by a cubical structure on the formal group law (Ando-Hopkins-Strickland 2001 Invent. Math. 146, 595-687 ^{[source pending]}).

The cubical-structure description is the algebraic content: an orientation $M U ⟨ 6 ⟩ \to E$ of an even-periodic ring spectrum $E$ corresponds to a trivialisation of $Θ^{3} (L)$ on the formal group $C$ of $E$ , where $L$ is the rigidified line bundle attached to the formal group and $Θ^{3}$ is Mumford's cubical-difference functor (theorem of the cube). For the universal elliptic curve over $M_{ell}$ , the existence of a cubical structure is the theorem of the cube; the Ando-Hopkins-Strickland theorem upgrades this existence statement to an $E_{\infty}$ -ring-spectrum map.

Theorem (Lurie derived-AG construction; Lurie 2009). The functor that assigns to a connective $E_{\infty}$ -ring $A$ the space of oriented derived elliptic curves over $A$ is representable by a derived Deligne-Mumford stack $M_{ell}^{der}$ whose underlying classical stack is the moduli stack of generalised elliptic curves; the global-sections spectrum is $TMF$ (Lurie 2009 Curr. Devel. Math. 2008, 219-277 ^{[source pending]}).

Lurie's representability theorem reduces the construction of $TMF$ to checking that a specific moduli problem in derived algebraic geometry is representable. The proof uses Artin-Lurie representability criteria for derived stacks together with explicit verification of formal smoothness and infinitesimal cohomological conditions for the oriented derived elliptic-curve moduli. The output identifies $TMF$ as the structure-sheaf global sections of a single derived geometric object.

Theorem (Stolz-Teichner conjecture). Conjecturally, there is a natural isomorphism $TMF^{n} (X) ≅ π_{0} (Hom ((1, n) -EFT, X))$ between the degree- $n$ topological modular forms of a manifold $X$ and concordance classes of $(1, n)$ -dimensional supersymmetric Euclidean field theories over $X$ (Stolz-Teichner 2011 AMS PSPM 83, 279-340 ^{[source pending]}).

The conjecture identifies $TMF$ as the classifying spectrum for two-dimensional supersymmetric Euclidean field theories, paralleling the (verified) identification of $K$ -theory with $1$ -dimensional supersymmetric Euclidean field theories. The conjecture is currently verified in degree zero on a point and in certain partial-degree statements; the full conjecture remains open and is one of the central open problems in the interface between mathematical physics and topology.

Synthesis. Quillen's theorem is the foundational reason that complex cobordism organises every complex-orientable cohomology theory: identifying $L = M U^{*} (pt)$ identifies the universal complex-oriented theory with the Lazard ring, and every other complex-orientable theory is exactly a base-change along a ring map $L \to R$ . The central insight is that the formal group law over $R$ is the only piece of data the resulting theory remembers. The chromatic-height filtration on formal group laws — height $0$ for the additive law, height $1$ for the multiplicative law, height $\geq 2$ for the formal group of an elliptic curve — then organises the cohomology-theoretic landscape: ordinary cohomology at height $0$ , $K$ -theory at height $1$ , elliptic cohomology at height $2$ . This is exactly the height-filtration structure that Devinatz-Hopkins-Smith 1988 ^{[source pending]} proved governs the nilpotence behaviour of the stable homotopy category.

Putting these together, the foundational mechanism that produces ordinary cohomology, $K$ -theory, and elliptic cohomology from complex cobordism is the same one: Landweber's exactness criterion converts a formal-group-law datum into an honest cohomology theory, and the topological-modular-form spectrum $TMF$ assembles the local elliptic spectra over the moduli stack of elliptic curves into a single global object. The bridge is the recognition that complex cobordism is universal among formal-group-theoretic cohomology theories; this pattern recurs in the construction of $TMF$ , where the moduli stack of elliptic curves replaces a single elliptic curve, and the Hopkins-Miller / Goerss-Hopkins / Ando-Hopkins-Strickland / Lurie machinery refines the underlying homotopy-ring-spectrum data to an $E_{\infty}$ -ring spectrum. The Witten genus then appears again in this framework as the $σ$ -orientation $MString \to TMF$ , identifying the modular-form-valued invariant of a string manifold with the image of its bordism class under a ring-spectrum map.

Full proof set Master

This is a pointer unit, so full proofs of the deep theorems above lie outside the scope of the present development; they would each require a dedicated unit of comparable length. The proofs below establish a self-contained proposition of the right flavour and indicate, for each major theorem cited in the Advanced results, the canonical primary reference where the full proof lives.

Proposition. The formal group law of complex $K$ -theory in the orientation $x = [L] - 1 \in K^{0} (CP^{\infty})$ , where $L$ is the tautological complex line bundle, is the multiplicative formal group law $F_{m} (X, Y) = X + Y + X Y$ .

Proof. The classifying space $CP^{\infty}$ classifies complex line bundles, and the multiplication map $μ : CP^{\infty} \times CP^{\infty} \to CP^{\infty}$ sends a pair of line bundles to their tensor product. Let $L_{1}, L_{2}$ be the pullbacks of the tautological line bundle along the two projections. Then $μ^{*} L = L_{1} \otimes L_{2}$ .

Apply $K^{0}$ to the multiplication map. The orientation pulls back as $$ \mu^* x = \mu^([L] - 1) = [\mu^ L] - 1 = [L_1 \otimes L_2] - 1. $$ On the other hand, $[L_{1} \otimes L_{2}] = [L_{1}] \cdot [L_{2}]$ in $K^{0}$ , so $$ [L_1 \otimes L_2] - 1 = [L_1][L_2] - 1 = ([L_1] - 1)([L_2] - 1) + ([L_1] - 1) + ([L_2] - 1). $$ Substituting $x_{i} = [L_{i}] - 1$ identifies this as $x_{1} x_{2} + x_{1} + x_{2} = F_{m} (x_{1}, x_{2})$ . The formal group law of complex $K$ -theory in this orientation is the multiplicative law. $□$

Proposition. The additive formal group law $F_{a} (X, Y) = X + Y$ over $Z$ does not satisfy the Landweber regularity criterion at any prime $p$ , so the assignment $X \mapsto MU_(X) \otimes_L \mathbb{Z}$ via the additive law is not given directly by Landweber's theorem.*

Proof. Under the Lazard-ring presentation, the additive law $F_{a}$ corresponds to the ring map $L \to Z$ sending every formal-group coefficient $a_{ij}$ to zero. The Hazewinkel generator $v_{n}$ at a prime $p$ has $p$ -adic expansion whose leading term is the height- $n$ coefficient of the $p$ -series $[p] (X)$ of the formal group law. For the additive law $F_{a}$ , the $p$ -series is $[p] (X) = pX$ , so $v_{n} = 0$ for every $n \geq 1$ when computed against $F_{a}$ . The Landweber criterion at the prime $p$ requires $v_{1}$ to be a non-zero-divisor in $Z / p$ ; since $v_{1} = 0$ in $Z$ , the criterion fails. (Ordinary integer cohomology still exists as the Eilenberg-MacLane spectrum $H Z$ , but it is not directly produced by the Landweber exact-functor theorem — Landweber's theorem produces the complex-orientable cohomology theories whose formal group is everywhere regular, and $H Z$ is the boundary case that requires a separate Brown-representability argument.) $□$

Theorem (Quillen). Stated above. Full proof: Quillen 1969 Bull. AMS 75, 1293-1298 ^{[source pending]}; expanded exposition Adams 1974 Stable Homotopy and Generalised Homology Part II ^{[source pending]}; Ravenel Complex Cobordism and the Stable Homotopy Groups of Spheres (Academic Press 1986) Ch. 4.

Theorem (Landweber). Stated above. Full proof: Landweber 1976 Amer. J. Math. 98, 591-610 ^{[source pending]}; modern exposition Ravenel 1986 Ch. 4, §4.2.

Theorem (Hopkins-Miller). Stated above. Full proof: Hopkins-Miller 1994-1999 manuscripts ^{[source pending]}; consolidated exposition Behrens et al. eds. 2014 Topological Modular Forms, AMS MSM 201 ^{[source pending]}.

Theorem (Ando-Hopkins-Strickland $σ$ -orientation). Stated above. Full proof: Ando-Hopkins-Strickland 2001 Invent. Math. 146, 595-687 ^{[source pending]}, where the cubical-structure identification and the construction of the $E_{\infty}$ -orientation are carried out in detail.

Theorem (Lurie derived-AG construction). Stated above. Full proof: Lurie 2009 Curr. Devel. Math. 2008, 219-277 ^{[source pending]}, with the representability of the oriented-derived-elliptic-curve moduli problem proved via Artin-Lurie representability criteria.

The cited primary sources contain the full arguments; the present unit indicates where each lives without repackaging it.

Connections Master

Modularity of the elliptic genus 03.06.23. The Witten genus, defined there as a multiplicative-sequence invariant valued in modular forms for $SL_{2} (Z)$ , refines to the Ando-Hopkins-Strickland $σ$ -orientation $MString \to TMF$ , identifying the modular-form-valued invariant with a topological-modular-form class. The elliptic-cohomology refinement explains why the Witten genus takes values in modular forms rather than in an arbitrary ring of formal power series: $TMF^{*} (pt)$ maps to the ring of integer modular forms, and the map becomes an isomorphism after inverting $6$ , so the Witten genus of a string manifold is the image of a topological-modular-form class under this comparison.
Multiplicative sequences and Hirzebruch genera 03.06.15. The classical Hirzebruch genera $L, \hat{A}, Td$ are all obtained as the genera attached to the additive (for $L, \hat{A}$ ) and multiplicative (for $Td$ ) formal group laws of ordinary cohomology and complex $K$ -theory respectively. The elliptic refinement places these genera inside a single chromatic tower: $L$ and $\hat{A}$ at height $0$ (ordinary cohomology), $Td$ at height $1$ ( $K$ -theory), the elliptic genera at height $2$ (elliptic cohomology). This is the foundational reason the classical genera fit into a hierarchy organised by formal group laws.
Oriented bordism and Pontryagin-Thom 03.06.13. The Pontryagin-Thom construction identifies the oriented bordism ring $Ω_{*}^{S O}$ with the homotopy groups of a Thom spectrum $MSO$ . The elliptic refinement upgrades the Thom-spectrum construction to a structured-ring-spectrum map: the bordism ring of string manifolds maps to $TMF^{*} (pt)$ via the $σ$ -orientation, and the multiplicative-sequence invariants of a bordism class are identified with the image of its Thom class in the topological-modular-form ring.
Topological K-theory 03.08.01. Complex $K$ -theory is the height- $1$ slot of the chromatic tower, sitting directly below elliptic cohomology at height $2$ . The formal group law of $K$ -theory is the multiplicative formal group law, equivalently the formal group of $G_{m}$ ; the formal group of an elliptic curve degenerates to $G_{m}$ at the nodal cubic, and this degeneration locus on the moduli stack of elliptic curves is where elliptic cohomology specialises to $K$ -theory. The relationship is the local-to-global story of the chromatic filtration in its first genuinely informative case.
Atiyah-Singer index theorem 03.09.10. The Witten genus heuristically arises as the $S^{1}$ -equivariant Dirac index on the free loop space of a string manifold; the Ando-Hopkins-Strickland refinement realises this heuristic as a rigorous map of ring spectra into $TMF$ . The classical Atiyah-Singer theorem identifies the $\hat{A}$ -genus with the index of the Dirac operator on a spin manifold; the elliptic refinement extends this identification: the Witten genus is the formal-loop-space analogue of the Dirac index, with the modular-form values reflecting the conformal symmetry of the loop space.
Modular forms on $SL_{2} (Z)$ 21.04.01. The ring of integer modular forms $MF_{*}$ is the target of the comparison map from $tmf_{*}$ , and the comparison becomes an isomorphism after inverting $6$ . Modular forms are therefore the arithmetic shadow of topological modular forms, and the topological refinement records the $2$ - and $3$ -torsion that the modular-form ring misses.

Historical & philosophical context Master

The story begins with Quillen's 1969 paper On the formal group laws of unoriented and complex cobordism theory (Bull. AMS 75, 1293-1298) ^{[Quillen 1969]}, which identified the formal group law of complex cobordism with the universal one-dimensional commutative formal group law and showed $L = M U^{*} (pt)$ . The result transformed algebraic topology by converting the classification of complex-orientable cohomology theories into the classification of formal group laws, and it placed the Lazard ring — which Lazard had introduced in 1955 in a purely algebraic context for classifying formal group laws — at the heart of stable homotopy theory.

Landweber's 1976 paper Homological properties of comodules over $MU_(MU) $an d$ BP_*(BP) $* (* A m er . J . M a t h . * 98, 591 - 610) [r e f : T O D O_{R} E F L an d w e b er 1976] p r o v i d e d t h ee x a c t - f u n c t or t h eor e m : a f l a t n esscr i t er i o n o na r in g ma p$ L \to R $e n s u r in g t ha t$ MU_*(-) \otimes_L R$ is a homology theory. This was the technical mechanism that allowed formal-group classifications to be promoted to cohomology theories. Applied to Weierstrass elliptic curves, the criterion produced the first explicit elliptic cohomology theories (Landweber-Ravenel-Stong 1995 AMS Cont. Math. 181, 317-337 ^{[Landweber 1988]}).

Ochanine 1987 and Witten 1987-1988 ^{[Witten 1988]} introduced the elliptic genera $φ_{Och}$ and $φ_{W}$ from a parallel direction motivated by mathematical physics, with Witten's loop-space heuristic identifying $φ_{W} (M)$ as the $S^{1}$ -equivariant Dirac index on the free loop space of a string manifold. Segal in his 1988 Séminaire Bourbaki exposé proposed that the elliptic genera should be invariants of an underlying generalised cohomology theory whose coefficient ring is a ring of modular forms — the elliptic-cohomology programme.

The next stage was the Hopkins-Miller construction of the topological-modular-form spectrum $TMF$ , developed through circulated manuscripts and seminars from 1994 to 1999 ^{[Hopkins Miller 1994]} and consolidated in the Behrens-Lawson-Hopkins-Miller volume Topological Modular Forms (AMS Math. Surveys & Monographs 201, 2014) ^{[Behrens 2014]}. The construction sheafified the local elliptic spectra over the moduli stack of elliptic curves and refined them to $E_{\infty}$ -ring spectra via Goerss-Hopkins obstruction theory (Goerss-Hopkins 2004 Lond. Math. Soc. LNS 315, 151-200 ^{[Goerss Hopkins 2004]}). Ando-Hopkins-Strickland 2001 (Invent. Math. 146, 595-687) ^{[Ando 2001]} then constructed the $σ$ -orientation $MString \to TMF$ , realising the Witten genus as a map of ring spectra and identifying its existence with a cubical structure on the formal group of the universal elliptic curve.

Lurie's 2009 survey A survey of elliptic cohomology (Curr. Devel. Math. 2008, 219-277) ^{[Lurie 2009]} reformulated the construction in derived algebraic geometry, identifying $TMF$ as the global sections of the structure sheaf of a derived Deligne-Mumford stack of oriented derived elliptic curves. The Lurie construction is conceptually streamlined: the existence of $TMF$ reduces to the representability of a moduli problem in derived algebraic geometry, sidestepping the obstruction-theoretic computations. Stolz-Teichner 2011 (Mathematical Foundations of Quantum Field Theory, AMS PSPM 83, 279-340) ^{[Stolz Teichner 2011]} proposed that $TMF$ classifies two-dimensional supersymmetric Euclidean field theories modulo concordance, paralleling the (known) identification of $K$ -theory with one-dimensional supersymmetric Euclidean field theories. The conjecture remains open in full generality but is verified in several partial cases.

Bibliography Master

@article{Quillen1969,
  author  = {Quillen, Daniel},
  title   = {On the formal group laws of unoriented and complex cobordism theory},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {75},
  year    = {1969},
  pages   = {1293--1298}
}

@article{Landweber1976,
  author  = {Landweber, Peter S.},
  title   = {Homological properties of comodules over $MU_*(MU)$ and $BP_*(BP)$},
  journal = {American Journal of Mathematics},
  volume  = {98},
  year    = {1976},
  pages   = {591--610}
}

@incollection{Landweber1988,
  author    = {Landweber, Peter S.},
  title     = {Elliptic cohomology and modular forms},
  booktitle = {Elliptic Curves and Modular Forms in Algebraic Topology},
  editor    = {Landweber, Peter S.},
  series    = {Lecture Notes in Mathematics},
  volume    = {1326},
  publisher = {Springer-Verlag},
  year      = {1988},
  pages     = {55--68}
}

@incollection{Witten1988,
  author    = {Witten, Edward},
  title     = {The index of the {D}irac operator in loop space},
  booktitle = {Elliptic Curves and Modular Forms in Algebraic Topology},
  editor    = {Landweber, Peter S.},
  series    = {Lecture Notes in Mathematics},
  volume    = {1326},
  publisher = {Springer-Verlag},
  year      = {1988},
  pages     = {161--181}
}

@incollection{LandweberRavenelStong1995,
  author    = {Landweber, Peter S. and Ravenel, Douglas C. and Stong, Robert E.},
  title     = {Periodic cohomology theories defined by elliptic curves},
  booktitle = {The {\v C}ech Centennial},
  editor    = {Cenkl, Mila and Miller, Haynes},
  series    = {Contemporary Mathematics},
  volume    = {181},
  publisher = {American Mathematical Society},
  year      = {1995},
  pages     = {317--337}
}

@article{AndoHopkinsStrickland2001,
  author  = {Ando, Matthew and Hopkins, Michael J. and Strickland, Neil P.},
  title   = {Elliptic spectra, the {W}itten genus, and the theorem of the cube},
  journal = {Inventiones Mathematicae},
  volume  = {146},
  year    = {2001},
  pages   = {595--687}
}

@inproceedings{Hopkins2002,
  author    = {Hopkins, Michael J.},
  title     = {Algebraic topology and modular forms},
  booktitle = {Proceedings of the International Congress of Mathematicians, Beijing 2002},
  volume    = {I},
  year      = {2002},
  pages     = {291--317}
}

@incollection{GoerssHopkins2004,
  author    = {Goerss, Paul G. and Hopkins, Michael J.},
  title     = {Moduli spaces of commutative ring spectra},
  booktitle = {Structured Ring Spectra},
  editor    = {Baker, Andrew and Richter, Birgit},
  series    = {London Mathematical Society Lecture Note Series},
  volume    = {315},
  publisher = {Cambridge University Press},
  year      = {2004},
  pages     = {151--200}
}

@incollection{Lurie2009,
  author    = {Lurie, Jacob},
  title     = {A survey of elliptic cohomology},
  booktitle = {Current Developments in Mathematics 2008},
  publisher = {International Press},
  address   = {Somerville, MA},
  year      = {2009},
  pages     = {219--277}
}

@incollection{StolzTeichner2011,
  author    = {Stolz, Stephan and Teichner, Peter},
  title     = {Supersymmetric field theories and generalized cohomology},
  booktitle = {Mathematical Foundations of Quantum Field Theory},
  editor    = {Sati, Hisham and Schreiber, Urs},
  series    = {Proceedings of Symposia in Pure Mathematics},
  volume    = {83},
  publisher = {American Mathematical Society},
  year      = {2011},
  pages     = {279--340}
}

@book{Behrens2014,
  editor    = {Behrens, Mark and Lawson, Tyler and Hopkins, Michael J. and Miller, Haynes R.},
  title     = {Topological Modular Forms},
  series    = {Mathematical Surveys and Monographs},
  volume    = {201},
  publisher = {American Mathematical Society},
  year      = {2014}
}

@book{Adams1974,
  author    = {Adams, J. Frank},
  title     = {Stable Homotopy and Generalised Homology},
  publisher = {University of Chicago Press},
  year      = {1974}
}

@article{DevinatzHopkinsSmith1988,
  author  = {Devinatz, Ethan S. and Hopkins, Michael J. and Smith, Jeffrey H.},
  title   = {Nilpotence and stable homotopy theory I},
  journal = {Annals of Mathematics},
  volume  = {128},
  year    = {1988},
  pages   = {207--241}
}

@book{Ravenel1986,
  author    = {Ravenel, Douglas C.},
  title     = {Complex Cobordism and the Stable Homotopy Groups of Spheres},
  publisher = {Academic Press},
  year      = {1986}
}

Prerequisites

03.06.23
03.06.15
03.06.13
03.08.01

Tier anchors

beginner: Lurie 2009 *A Survey of Elliptic Cohomology* introduction; Landweber 1988 *LNM* 1326 preface
intermediate: Lurie 2009 *A Survey of Elliptic Cohomology* (Curr. Devel. Math. 2008, 219-277); Landweber 1988 *LNM* 1326, 55-68; Hopkins 2002 ICM Vol. I, 291-317
master: Quillen 1969 *Bull. AMS* 75, 1293-1298; Landweber 1976 *Amer. J. Math.* 98, 591-610; Landweber-Ravenel-Stong 1995 *AMS Cont. Math.* 181, 317-337; Goerss-Hopkins 2004 *Lond. Math. Soc. LNS* 315, 151-200; Ando-Hopkins-Strickland 2001 *Invent. Math.* 146, 595-687; Hopkins 2002 ICM Vol. I, 291-317; Lurie 2009 *Curr. Devel. Math.* 2008, 219-277; Stolz-Teichner 2011 AMS PSPM 83, 279-340; Behrens et al. eds. 2014 *Topological Modular Forms*, AMS Math. Surveys & Monographs 201

References

Quillen, D. — On the formal group laws of unoriented and complex cobordism theory · *Bull. Amer. Math. Soc.* 75 (1969), 1293-1298. Originator paper. Identifies the formal group law of $MU^*$ with the universal one-dimensional commutative formal group law over the Lazard ring, and proves $L \cong MU^*(\mathrm{pt})$ as graded rings.
Landweber, P. S. — Homological properties of comodules over $MU_*(MU)$ and $BP_*(BP)$ · *Amer. J. Math.* 98 (1976), 591-610. The Landweber exact-functor theorem: a flatness criterion for a ring map $MU_* \to R$ that ensures $MU_*(-) \otimes_{MU_*} R$ is a homology theory.
Landweber, P. S. (ed.) — Elliptic Curves and Modular Forms in Algebraic Topology · Lecture Notes in Mathematics 1326, Springer 1988. Proceedings of the Princeton 1986 workshop; contains Landweber's survey 'Elliptic cohomology and modular forms' (pp. 55-68), Witten's 'The index of the Dirac operator in loop space' (pp. 161-181), and Zagier's 'Note on the Landweber-Stong elliptic genus' (pp. 216-224).
Landweber, P. S., Ravenel, D. C., Stong, R. E. — Periodic cohomology theories defined by elliptic curves · in *The Čech Centennial* (M. Cenkl, H. Miller, eds.), *AMS Cont. Math.* 181 (1995), 317-337. Period-map construction of elliptic cohomology theories from Weierstrass elliptic curves over $\mathbb{Z}[1/6]$; explicit Landweber-exactness verification.
Hopkins, M. J., Miller, H. R. — Elliptic cohomology I, II (manuscripts 1994-1999) · Circulated MIT manuscripts (Hopkins-Miller 1994-1999) published in expanded form in M. Behrens, T. Lawson, M. J. Hopkins, H. R. Miller, et al. (eds.), *Topological Modular Forms*, AMS Mathematical Surveys & Monographs 201 (2014). Construction of the $\mathrm{TMF}$ ring spectrum via Goerss-Hopkins obstruction theory on the moduli stack of elliptic curves.
Goerss, P. G., Hopkins, M. J. — Moduli spaces of commutative ring spectra · in *Structured Ring Spectra* (A. Baker, B. Richter, eds.), London Math. Soc. Lecture Note Series 315, Cambridge Univ. Press 2004, 151-200. Foundational paper introducing the obstruction-theoretic framework for refining a homotopy ring spectrum to an $E_\infty$-ring spectrum, with obstructions in André-Quillen cohomology.
Ando, M., Hopkins, M. J., Strickland, N. P. — Elliptic spectra, the Witten genus, and the theorem of the cube · *Invent. Math.* 146 (2001), 595-687. Construction of the $\sigma$-orientation $\mathrm{MString} \to \mathrm{TMF}$ realising the Witten genus as a ring-spectrum map; identification of elliptic cohomology orientations with cubical structures on the formal group of the elliptic curve.
Hopkins, M. J. — Algebraic topology and modular forms · Proc. ICM 2002, Beijing, Vol. I, 291-317. ICM exposition of $\mathrm{TMF}$; the comparison map $\mathrm{tmf}_* \to \mathrm{MF}_*$ to integer modular forms and the isomorphism after inverting $6$.
Lurie, J. — A survey of elliptic cohomology · *Current Developments in Mathematics* 2008 (Int. Press, Somerville, MA, 2009), 219-277. Derived-algebraic-geometry construction of $\mathrm{TMF}$ as global sections of a sheaf of $E_\infty$-rings on the moduli stack of oriented derived elliptic curves; chromatic-height perspective on Morava $E$-theories.
Stolz, S., Teichner, P. — Supersymmetric field theories and generalized cohomology · in *Mathematical Foundations of Quantum Field Theory* (H. Sati, U. Schreiber, eds.), AMS Proc. Symp. Pure Math. 83 (2011), 279-340. Conjectural identification of $\mathrm{TMF}^n(X)$ with concordance classes of $(1, n)$-dimensional supersymmetric Euclidean field theories over $X$.
Witten, E. — The index of the Dirac operator in loop space · in *Elliptic Curves and Modular Forms in Algebraic Topology* (P. S. Landweber, ed.), Lecture Notes in Mathematics 1326, Springer 1988, 161-181. Originator paper for the Witten genus on string manifolds; loop-space heuristic identifying $\varphi_W(M)$ with the $S^1$-equivariant Dirac index on the free loop space $\mathcal{L}M$.
Adams, J. F. — Stable Homotopy and Generalised Homology · University of Chicago Press, Chicago Lectures in Mathematics, 1974. Standard reference for the calculation of $MU_*$ and $MU^*(MU)$, the role of formal group laws, and Brown-Peterson spectra $BP$ at a prime $p$.
Behrens, M., Lawson, T., Hopkins, M. J., Miller, H. R., et al. (eds.) — Topological Modular Forms · AMS Mathematical Surveys & Monographs 201 (2014). Edited collection assembling the Hopkins-Miller construction, the Goerss-Hopkins obstruction theory, the computation of $\mathrm{tmf}_*$, and the comparison with integer modular forms; the definitive reference.
Devinatz, E. S., Hopkins, M. J., Smith, J. H. — Nilpotence and stable homotopy theory I · *Annals of Mathematics* 128 (1988), 207-241. Nilpotence theorem and the chromatic-height filtration of the stable homotopy category; foundational for the height-$n$ Morava picture in which elliptic cohomology sits at height $2$.
Hatcher, A. — Algebraic Topology · Cambridge University Press 2002. Standard textbook reference for ordinary singular cohomology, $K$-theory background, and the generalised-cohomology axioms underlying the Eilenberg-Steenrod / Brown representability framework on which the elliptic refinement is built.

Estimated time

beginner: 12m
intermediate: 30m
master: 70m