Talbot 2007: Topological Modular Forms

Mentored by Mike Hopkins.

March 25 to March 31, 2007.
North Conway, New Hampshire.


Here is the homepage for the book based on the 2007 Talbot workshop.

Talk Schedule

Monday Tueseday Wednesday Thursday Friday

Historic Overview

The Landweber exact functor theorem

The Hasse square

Goerss-Hopkins obstruction theory

$K(1)$-local obstruction theory

Elliptic curves and Modular forms

$E_\infty$ ring spectra

Model categories

The Hopkins-Miller theorem

The construction of $TMF$

Algebraic stacks

Sheaves in homotopy theory

The big spectral sequence [Mike Hopkins]

The string orientation [Mike Hopkins]


The geometry of $M_{Ell}$

The TMF theorem [Mike Hopkins]

Universal deformations

discussion session

Title to be announced [Mike Hopkins]

-- Monday --

1. Historic Overview. [Corbett]
Introduce general (co)homology theories and the examples of $K$-theory and complex cobordism. Define a genus as a multiplicative invariant of almost complex manifolds and give the example of the Todd Genus. State Quillen's theorem and establish the equivalence between genera with values in $R$, and formal group laws (FGL) over $R$, under certain conditions, given a genus $g:MU_* \to R$, one can find a homology theory $E_*(-)$ with $E_*=R$, represented by a spectrum $E$, and a map of ring spectra $MU \to E$ realizing $g$ at the level of homotopy. Define elliptic cohomology.

2. Elliptic curves and Modular forms. [Carl]
Introduce elliptic curves as cubic curves in $\mathbb{P}^2$, and as one dimensional abelian group schemes. Give a hint of why the two definitions are equivalent. Give some examples of elliptic curves. Define the canonical line bundle ω over the modui space of elliptic curves as an assignment {elliptic curve over some field} $\mapsto$ {one dimensional vector space over that field}. Define modular forms as sections of $\omega^n$ and give some examples of them. Describe the ring of integral modular forms.

3. Algebraic stacks. [Nick]
Explain the formalism of functor of points and introduce general stacks. Give the examples of the moduli space of elliptic curves $M_{Ell}$ and the moduli space of formal groups $M_{FG}$. Explain what a groupoid object in schemes is and how it represents a stack. Work out the groupoid representing $M_{Ell}$. Explain what a line bundle over a stack is and construct $\omega$ rigorously. Show that $\omega$ is pulled back from a map $M_{Ell}\to M_{FG}$.

4. The geometry of $M_{Ell}$. [Andre]
Introduce the notions of $p$-typical formal group law and explain why every formal group law over a field can be isomorphed to a $p$-typical one. State the classification of formal groups over algebraically closed fields. Introduce the notions of ordinary versus supersingular elliptic curves and give some examples at the primes 2 and 3. The stack of supersingular elliptic curves $M_{Ell}^{\mathit{ss}}$ is a zero dimensional substack of $M_{Ell}\otimes \Z/p$, and $M_{Ell}^{\mathit{ord}}$ is its open complement. The stack $M_{Ell}^{\mathit{ord}}$ still makes sense as an open substack of $M_{Ell} \otimes \Z_p$. Explain why $M_{Ell}^{\mathit{ss}}$ is not well defined as a substack of $M_{Ell}\otimes \Z_p$, but its formal neighborhood still makes sense.

-- Tuesday --

5. The Landweber exact functor theorem. [Henning]
Explain how one assigns a FG to an even periodic cohomology theory. Introduce $MU$ and its periodic version $MP$. State Quillen's theorem and explain why the groupoid $\mathrm{Spec}(MP_0MP)\to \mathrm{Spec}(MP_0)$ represents $M_{FG}$. State the Landweber exact functor theorem (LEFT). Define what it means for a map from $\mathrm{Spec}(R)$ to a stack to be flat. Explain why the statement of LEFT is equivalent to the following statement: If a map $MP_0\to R$ induces a flat map $\mathrm{Spec}(R) \to M_{FG}$, then the functor $X \to MP_{*}(X)\otimes_{P_0} R$ is a homology theory. Prove that the map $M_{Ell} \to M_{FG}$ is flat and use it to build a presheaf {flat maps from an affine scheme to $M_{Ell}$} $\to$ {even periodic homology theories}.

6. $E_\infty$ ring spectra. [Matthew]
Recall what a spectrum is and introduce the formalism of symmetric spectra. Explain what operads are, and give the examples of the $A_\infty$ and of the $E_\infty$ operads. Give two possible definitions of $E_\infty$ ring spectra: as spectra with a multiplication which is commutative up to infinitely many homotopies i.e. which is equipped with an action of an $E_\infty$ operad, and as a spectrum with a strictly commutative multiplication. Give a hint of why those two approaches are equivalent. Explain the philosophy of why {(co)homology theories} = {spectra} but {cohomology theories equipped with a commutative product} $\neq$ {$E_\infty$ ring spectra}.

7. Sheaves in homotopy theory. [Chris]
Explain what a sheaf on a stack is, and why it's not that different from a sheaf on a space. Define sheaf cohomology. If a presheaf takes values in a category in which homotopy limits make sense, then one can modify the descent condition to include higher homotopies on the $n$-tuple intersections: this is the homotopy meaningful replacement of the concept of sheaf. Setup the spectral sequence (SS) which, given a sheaf of spectra $F$ over $X$ computes the homotopy groups of the spectrum of global section π_{*}($F$($X)$). Its $E$_{2} term is the sheaf cohomology of the sheafification of the presheaf $U$ |\to π_{*}($F$($U)$). A special case of this SS will be used in talk 11 to compute π_{*}($F)$ .

8. The $F$ theorem. [Mike Hopkins]
This talk contains a survey of the second half of the workshop. The main theorem is about the existence of a sheaf of $E_\infty$ ring spectra $O^{top}$ over $M_{Ell}$ that recovers the presheaf of talk 5 at the level of the corresponding homology theories. The spectrum of global sections of that sheaf is $F$.

-- Wednesday --

9. The Hasse square. [Tilman]
Explain what it means to localize a spectrum with respect to a cohomology theory and give the examples of rationalization and $p$-completion. State Sullivan's arithmetic square. Define the Morava $K$-theories $K$($n)$ and the corresponding localization functors $L$_{$K$($n)$}. When applied to Landweber exact cohomology theories, $L$_{$K$($n)$} kills everything that is not of height $n$. When applied to the sheaf of spectra $O^{top}$ over $M_{Ell}$, explain why $L$_{$K$(2)} = (completion around the supersingular locus) and $L$_{$K$(1)} = (localization away from the supersingular locus). Prove that, given a $K$(1)v$K$(2)-local spectrum $X$, the Hasse square $L$_{$K$(2)} $X$ // $L$_{$K$(1)} $L$_{$K$(1)} $L$_{$K$(2)} $X$ is a homotopy pullback.

10. Model categories. [Andrew]
Introduce the formalism of cofibrantly generated model categories. Explain the notions of fibrant and cofibrant objects. Describe the model structure on the category of $E_\infty$ ring spectra. Given a model category $M$ and a space (or stack) $X$, describe the Jardine model category structure on the category of sheaves on $X$ with values in $M$. Show that the fibrant objects in that model category are sheaves in the sense of talk 7.

11. The big spectral sequence. [Mike Hopkins]
Mike will explain what the big spectral sequence $H^{s}(M_{Ell}\omega^{t}) \Rightarrow \pi_s(F)$ looks like, and give some applications.

12. Universal deformations. [Jacob]
Explain what it means to be a universal deformations of a FGL over a field $k$, and why it's equivalent to taking formal neighborhoods of points inside the stack $M_{FG}$. Give Lubin-Tate's explicit description of the universal deformation of a FGL. Define and describe the Morava $E$-theories $E_{n}$. Prove the Serre-Tate theorem according to which the formal neighborhood in $M_{Ell}$ of a supersingular curve is isomorphic to the formal neighborhood of the corresponding point in $M_{FG}$.

-- Thursday --

13. Goerss-Hopkins obstruction theory. [Vigleik]
The Goerss-Hopkins obstruction theory tries to answer the following question: Suppose $E_{*}(-)$ is a "good" homology theory and we have an $E_{*}$-algebra $A$ in the cateogry of $E_{*}E$ comodules ("good" means that spectrum $E$ has a homotopy commutative multiplication, that $E_{*}E$ is flat over $E_{*}$, and that the Adams condition is satisfied). Is there an $E_\infty$ ring spectrum $X$ with $E_{*}X=A$? More precisely, what is the moduli space of such $E_\infty$ ring spectra? To answer this question, one defines a new type of homotopy groups which have a good theory of Postnikov towers and of $k$-invariants. One then tries to build $X$ one Postnikov section at a time, and identify the obstructions to the existence of the right $k$-invariants. The answer turns out to lie in certain Andre-Quillen cohomology groups. This theory can also be used to compute the homotopy groups of the space of $E_\infty$ maps between two $E_\infty$ ring spectra. In talks 14 and 16, this theory will be specialized to the cases $E=E_{n}$, and $E=K(1)$ respectively.

14. The Hopkin-Miller theorem. [John F.]
Given a formal group law over a field, its universal deformation is always Landweber exact, i.e. the corresponding map $\mathrm{Spec}(R) \to M_{FG}$ is flat. Thus, we get a functor {formal group laws over fields} $\to$ {even periodic homology theories}, by first taking the universal deformation, and then applying LEFT. The Hopkins-Miller theorem says that one can lift that functor to the category of $E_\infty$ ring spectra. The existence of such a lift is proven by considering all those $E_\infty$ ring spectra which could possibly be in the image of the functor. They form a category, which is naturally equipped with a functor to the category of FG's over fields. Using the GH obstruction theory, one shows that that functor is a homotopy equivalence, and thus that it has an inverse. We then apply the Hopkins-Miller theorem to construct $L_{K(2)} O^{top}$ and $L_{K(2)} F$.

15. The string orientation. [Mike Hopkins]
In this talk, Mike will introduce Rezk's logarithm and construct the $O\langle 8\rangle$-orientation, which is a map of $E_\infty$ ring spectra $O\langle 8\rangle \to F$.

-- Friday --

16. $K(1)$-local obstruction theory. [Mike Hill]
When working inside the cateogry of $K(1)$-local spectra, the appropriate variant of GH obstruction theory is the one based on $K(1)$. This has the advantage that the algebra playing the role of the Dyer-Lashof algebra is very small: it's generated by a single operation $\theta : x \mapsto (\psi_{p}(x) - x^{p}) / p$. So our variant of GH obstruction theory now looks as follows: it starts with a $K(1)_{*}K(1)$-comodule algebra, and it tells us whether there exists a $K(1)$-local $E_\infty$ ring spectrum $X$, whose $K(1)$-homology $K(1)_{*}X$ has the above structure.

17. The construction of $F$. [Mark]
Even though we don't know yet what $O^{top}$ is, we know what $K(1)_{*}(O^{top})$ should look like, as a sheaf of $K(1)_{*}K(1)$-comodule algebras. Applying the GH obstruction theory in the category of (pre)sheaves of $K(1)$-local spectra, one shows that there's a unique possible value for $L_{K(1)} (O^{top})$. Similarly, one can compute $K(1)_{*} (L_K(2)) O^{top})$ as a sheaf of $K(1)_{*}K(1)$-comodule algebras. Applying again the GH obstruction theory, one checks that there's a unique possible map $L_{K(1)} O^{top}\to L_{K(2)} O^{top}$. One then defines the $p$-completion of $O^{top}$ to be the homotopy pullback of $L_{K(1)} O^{top} \to L_{K(1)} L_{K(2)} O^{top}\leftarrow L_{K(2)} O^{top}$. Finally, one uses Sullivan's arithmetic square to construct the global version of $O^{top}$.

18. Title to be announced. [Mike Hopkins]
In this talk, Mike could talk about the fiber of the logarithm, relations with number theory, things related to the work of Mazur and Wiles, or other stuff...