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…)