In mathematics, the moduli stack of elliptic curves, denoted as M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} or M e l l {\displaystyle {\mathcal {M}}_{\mathrm...

a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic...

and the moduli stack of elliptic curves. Originally, they were introduced by Alexander Grothendieck to keep track of automorphisms on moduli spaces, a...

constructing moduli spaces as algebraic stacks from moduli functors Moduli of algebraic curves Moduli stack of elliptic curves Moduli spaces of K-stable Fano...

\Gamma ={\text{SL}}_{2}(\mathbb {Z} )} are sections of a line bundle on the moduli stack of elliptic curves. A modular function is a function that is invariant...

geometry) Modularity theorem Moduli stack of elliptic curves Nagell–Lutz theorem Riemann–Hurwitz formula Wiles's proof of Fermat's Last Theorem Sarli,...

constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations of situations...

generalization of elliptic curves to higher dimensions. However, unlike the case of elliptic curves, there is no well-behaved stack playing the role of a moduli stack...

m_{n}} is the multiplication by n. See also: modular curve#Examples, moduli stack of elliptic curves. Siegel modular form Rigidity (mathematics) Local rigidity...

{Spectra}}} over the site of affine schemes flat over the moduli stack of elliptic curves. The desire to get a universal elliptic cohomology theory by taking...

the moduli stack of (generalized) elliptic curves. This theory has relations to the theory of modular forms in number theory, the homotopy groups of spheres...

stacks. In particular, these conditions are used in the construction of the moduli stack of elliptic curves and the construction of the moduli stack of...

after many years of effort. Level structure (algebraic geometry) Moduli stack of elliptic curves Drinfeld, Vladimir (1974), "Elliptic modules", Matematicheskii...

In the study of the arithmetic of elliptic curves, the j-line over a ring R is the coarse moduli scheme attached to the moduli problem sending a ring...

completed his PhD in 1961, with a thesis entitled Existence of the moduli scheme for curves of any genus. Mumford's work in geometry combined traditional geometric...

Fundamental domain Half-space Kleinian group Modular group Moduli stack of elliptic curves Riemann surface Schwarz–Ahlfors–Pick theorem Weisstein, Eric...

structures, all of the form C / (Z + τZ), where τ is any complex non-real number. These are called elliptic curves. Important examples of non-compact Riemann...

1\mod N,c\equiv 0\mod N\right\}.} These curves have a direct interpretation as moduli spaces for elliptic curves with level structure and for this reason...

idea that the moduli problem is to express the algebraic structure naturally coming with a set (say of isomorphism classes of elliptic curves). The result...

which are a family of elliptic curves degenerating to a rational curve with a cusp. One of the most important properties of stable curves is the fact that...

natural moduli problem or, in the precise language, there is no natural moduli stack that would be an analog of moduli stack of stable curves. An algebraic...

Differential of the first kind Jacobian variety Generalized Jacobian Moduli of algebraic curves Hurwitz's theorem on automorphisms of a curve Clifford's...

rational curves, i.e. the curve is birational to the projective line P 1 {\displaystyle \mathbb {P} ^{1}} . (b) g = 1 {\displaystyle g=1} . Elliptic curves, i...

hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies...

the first-order deformation space of X. This is the basic calculation needed to show that the moduli space of curves of genus g has dimension 3 g − 3 {\displaystyle...

is Zariski dense there). All elliptic curves are the Jacobian of themselves, hence the moduli stack of elliptic curves M 1 , 1 {\displaystyle {\mathcal...

Chapter II. Stacks Project, Tag 020D. Hartshorne 1997, Proposition II.2.3. Eisenbud & Harris 1998, Proposition VI-2. "Elliptic curves" (PDF). p. 20...

representations of the étale fundamental group of an algebraic curve to objects of the derived category of l-adic sheaves on the moduli stack of vector bundles...

the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite...

work also gave rise to the ideas of an algebraic space and algebraic stack, and has proved very influential in moduli theory. He also has made important...