I’ve just uploaded to arXiv my paper “The homology of ${\mathrm{tmf}}$,” which is an outgrowth of a project I was working on last summer. The main result of the paper is a description, well-known in the field but never written down in detail, of the mod ${2}$ cohomology of the spectrum ${\mathrm{tmf}}$ of (connective) topological modular forms, as a module over the Steenrod algebra: one has

$\displaystyle H^*(\mathrm{tmf}; \mathbb{Z}/2) \simeq \mathcal{A} \otimes_{\mathcal{A}(2)} \mathbb{Z}/2,$

where ${\mathcal{A}}$ is the Steenrod algebra and ${\mathcal{A}(2) \subset \mathcal{A}}$ is the 64-dimensional subalgebra generated by ${\mathrm{Sq}^1, \mathrm{Sq}^2,}$ and ${ \mathrm{Sq}^4}$. This computation means that the Adams spectral sequence can be used to compute the homotopy groups of ${\mathrm{tmf}}$; one has a spectral sequence

$\displaystyle \mathrm{Ext}^{s,t}( \mathcal{A} \otimes_{\mathcal{A}(2)} \mathbb{Z}/2, \mathbb{Z}/2) \simeq \mathrm{Ext}^{s,t}_{\mathcal{A}(2)}(\mathbb{Z}/2, \mathbb{Z}/2) \implies \pi_{t-s} \mathrm{tmf} \otimes \widehat{\mathbb{Z}_2}.$

Since ${\mathcal{A}(2) \subset \mathcal{A}}$ is finite-dimensional, the entire ${E_2}$ page of the ASS can be computed, although the result is quite complicated. Christian Nassau has developed software to do these calculations, and a picture of the ${E_2}$ page for ${\mathrm{tmf}}$ is in the notes from André Henriques‘s 2007 talk at the Talbot workshop. (Of course, the determination of the differentials remains.)

The approach to the calculation of ${H^*(\mathrm{tmf}; \mathbb{Z}/2)}$ in this paper is based on a certain eight-cell (2-local) complex ${DA(1)}$, with the property that

$\displaystyle \mathrm{tmf} \wedge DA(1) \simeq BP\left \langle 2\right\rangle,$

where ${BP\left \langle 2\right\rangle = BP/(v_3, v_4, \dots, )}$ is a quotient of the classical Brown-Peterson spectrum by a regular sequence. The usefulness of this equivalence, a folk theorem that is proved in the paper, is that the spectrum ${BP\left \langle 2\right\rangle}$ is a complex-orientable ring spectrum, so that computations with it (instead of ${\mathrm{tmf}}$) become much simpler. In particular, one can compute the cohomology of ${BP\left \langle 2\right\rangle}$ (e.g., from the cohomology of ${BP}$), and one finds that it is cyclic over the Steenrod algebra. One can then try to “descend” to the cohomology of ${\mathrm{tmf}}$. This “descent” procedure is made much simpler by a battery of techniques from Hopf algebra theory: the cohomologies in question are graded, connected Hopf algebras. (more…)

Let ${M_{1, 1}}$ be the moduli stack of elliptic curves. Given a scheme ${S}$, maps ${S \rightarrow M_{1, 1}}$ are given by the groupoid of elliptic curves over ${S}$, together with isomorphisms between them. The goal of this post is to compute ${\mathrm{Pic}(M_{1, 1})}$ away from the primes ${2, 3}$. (This is done in Mumford’s paper “Picard groups of moduli problems.”)

In the previous post, we saw that ${M_{1, 1}}$ could be described as a quotient stack. Namely, consider the scheme ${B_1 = \mathrm{Spec} \mathbb{Z}[a_1, a_2, a_3, a_4, a_6]}$ and the Weierstrass equation

$\displaystyle Y^2 Z + a_1 XYZ + a_3 YZ^2 = X^3 + a_2 X^2 Z + a_4 XZ^2 + a_6 Z^3$

cutting out a subscheme ${E_1 \subset \mathbb{P}^2_{B_1}}$. This is a flat family of projective cubic curves over ${\mathbb{P}^2_{B_1}}$ with a section (the point at infinity given by ${[X: Y: Z] = [0 : 1 : 0]}$). There is an open subscheme ${B \subset B_1}$ over which the family ${E_1 \rightarrow B_1}$ is smooth, i.e., consists of elliptic curves. A little effort with cohomology and Riemann-Roch allows us to show that, Zariski locally, any elliptic curve ${X \rightarrow S}$ can be pulled back from one of these: that is, any elliptic curve locally admits a Weierstrass equation.

The Weierstrass equation was not unique, though; any change of parametrization (in affine coordinates here)

$\displaystyle x' = a^2 x + b, \quad y' = a^3 x + c + d, \ a \mathrm{\ invertible}$

preserves the form of the equation, and these are the only transformations preserving it. In other words, the map

$\displaystyle B \rightarrow M_{1, 1}$

exhibits ${B}$ as a torsor over ${M_{1,1}}$ for the group scheme ${\mathbb{G} = \mathrm{Spec} \mathbb{Z}[a^{\pm 1}, b, c, d]}$ with a multiplication law given by composing linear transformations. That is,

$\displaystyle M_{1, 1} \simeq B/\mathbb{G};$

that is, to give a map ${S \rightarrow M_{1, 1}}$, one has to choose an étale cover ${\left\{S_\alpha\right\}}$ of ${S}$ (Zariski is enough here), maps ${S_\alpha \rightarrow B}$ inducing elliptic curves over the ${S_\alpha}$, and isomorphisms (coming from maps to ${\mathbb{G}}$) over ${S_\alpha \times_S S_\beta}$. (more…)

Let ${S}$ be a scheme. An elliptic curve over ${S}$ should be thought of as a continuously varying family of elliptic curves parametrized by ${S}$.

Definition 1 An elliptic curve over ${S}$ is a proper, flat morphism ${p: X \rightarrow S}$ whose geometric fibers are curves of genus one together with a section ${0: S \rightarrow X}$.

This is a reasonable notion of “family”: observe that a morphism ${T \rightarrow S}$ can be used to pull back elliptic curves over ${S}$. The flatness condition can be thought of as “continuity.” For an algebraically closed field, this reduces to the usual notion of an elliptic curve.

A basic property of elliptic curves over algebraically closed fields is that they imbed into ${\mathbb{P}^2}$ and are cut out by (nonsingular) Weierstrass equations of the form

$\displaystyle Y^2 Z + a_1 XYZ + a_3 YZ^2 = X^3 + a_2 X^2 Z + a_4 XZ^2 + a_6 Z^3.$

This equation is unique up to an action of a certain four-dimensional group of transformations. The first goal is to show that, locally, the same is true for an elliptic curve over a base.  (more…)