Moduli stack of elliptic curves

In mathematics, the moduli stack of elliptic curves, denoted as ${\displaystyle {\mathcal {M}}_{1,1}}$ or ${\displaystyle {\mathcal {M}}_{\mathrm {ell} }}$, is an algebraic stack over ${\displaystyle {\text{Spec}}(\mathbb {Z} )}$ classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves ${\displaystyle {\mathcal {M}}_{g,n}}$. In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme ${\displaystyle S}$ to it correspond to elliptic curves over ${\displaystyle S}$. The construction of this space spans over a century because of the various generalizations of elliptic curves as the field has developed. All of these generalizations are contained in ${\displaystyle {\mathcal {M}}_{1,1}}$.