Elliptic Curves

Let \(M_\ell\) denote the Deligne-Mumford stack of Elliptic Curves over \(\operatorname{Spec}({\mathbb{Z}})\). See also moduli stack of elliptic curves.

The stack \(M_{\ell}\) admits a compactification \({ \mathcal{M}_{\mathrm{ell}} }\) whose \(R\) points are generalized Elliptic Curves. The space of integral modular forms of weight \(k\) is defined to be the sections \begin{align*} H ^ { 0 } \left( \mkern 1.5mu\overline{\mkern-1.5mu{ \mathcal{M}_{\mathrm{ell}} }\mkern-1.5mu}\mkern 1.5mu , \omega{ {}^{ \scriptscriptstyle\otimes_{k}^{n} } } \right) \end{align*} Motivated by the definition of integral modular forms and this descent spectral sequence in the case of \(U = M_\ell\) , the spectrum $\mathrm{TMF} $ is defined to be the global sections \begin{align*} \mathrm{TMF} \coloneqq{\mathcal{O}}^ { {\mathsf{Top}}} \left( \mkern 1.5mu\overline{\mkern-1.5mu{ \mathcal{M}_{\mathrm{ell}} }\mkern-1.5mu}\mkern 1.5mu \right) \end{align*}

Interesting Space: Elliptic Curves