In the study of the arithmetic of

elliptic curves In mathematics, an elliptic curve is a Smoothness, smooth, Projective variety, projective, algebraic curve of Genus of an algebraic curve, genus one, on which there is a specified point . An elliptic curve is defined over a field (mathematics), ...

, the ''j''-line over a

ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

''R'' is the coarse

moduli scheme In algebraic geometry, a moduli scheme is a moduli space that exists in the category of schemes developed by French mathematician Alexander Grothendieck. Some important moduli problems of algebraic geometry can be satisfactorily solved by means ...

attached to the moduli problem sending a ring

R

to the set of isomorphism classes of elliptic curves over

R

. Since elliptic curves over the complex numbers are isomorphic (over an algebraic closure) if and only if their

j

-invariants agree, the affine space

\mathbb^1_j

parameterizing j-invariants of elliptic curves yields a

coarse moduli space In algebraic geometry, a moduli scheme is a moduli space that exists in the category of schemes developed by French mathematician Alexander Grothendieck. Some important moduli problems of algebraic geometry can be satisfactorily solved by means of ...

. However, this fails to be a fine moduli space due to the presence of elliptic curves with automorphisms, necessitating the construction of the

Moduli stack of elliptic curves In mathematics, the moduli stack of elliptic curves, denoted as \mathcal_ or \mathcal_, is an algebraic stack over \text(\mathbb) classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves \mathcal_. In part ...

. This is related to the

congruence subgroup In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of invertible integer matrices of determinant 1 in which the off-diag ...

\Gamma(1)

in the following way: :

M(Gamma(1) = \mathrm(R

Here the ''j''-invariant is normalized such that

j=0

has

complex multiplication In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...

\mathbb zeta_3 /math>, and j=1728 has complex multiplication by \mathbb /math>.

The ''j''-line can be seen as giving a coordinatization of the

classical modular curve In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation :, such that is a point on the curve. Here denotes the -invariant. The curve is sometimes called , though often that notation is used f ...

of level 1,

X_0(1)

, which is isomorphic to the

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

projective line In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ...

\mathbb^1_

.. See in particula
p. 378

References

Moduli theory Elliptic curves {{math-stub