Editing Modular curve (section)

{{Short description|Algebraic variety}}
In [[number theory]] and [[algebraic geometry]], a '''modular curve''' ''Y''(Γ) is a [[Riemann surface]], or the corresponding [[algebraic curve]], constructed as a [[Quotient by a group action|quotient]] of the complex [[upper half-plane]] '''H''' by the [[Group action (mathematics)|action]] of a [[congruence subgroup]] Γ of the [[modular group]] of integral 2×2 matrices SL(2,&nbsp;'''Z'''). The term modular curve can also be used to refer to the '''compactified modular curves''' ''X''(Γ) which are [[compactification (mathematics)|compactification]]s obtained by adding finitely many points (called the '''cusps of Γ''') to this quotient (via an action on the '''extended complex upper-half plane'''). The points of a modular curve [[moduli problem|parametrize]] isomorphism classes of [[elliptic curve]]s, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to [[complex number]]s, and, moreover, prove that modular curves are [[field of definition|defined]] either over the field of [[rational number]]s '''Q''' or a [[cyclotomic field]] '''Q'''(ζ<sub>''n''</sub>). The latter fact and its generalizations are of fundamental importance in number theory.