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ANALYTIC DEFINITION AND EXAMPLES The Modular Group acts on the upper half-plane by Fractional Linear Transformation s. The analytic definition of a modular curve involves a choice of a subgroup Γ satisfying an important technical condition. For any ''N'' ≥ 1, Γ(''N'') denotes the subgroup of the modular group consisting of matrices that are congruent to the identity matrix '' Modulo '' ''N'', it is called the principal congruence subgroup of level ''N'' . The technical condition is that Γ must contain the subgroup Γ(''N'') for some ''N'', and the minimal such ''N'' is called the '''level''' of Γ. Then the modular curve ''X''(Γ) is the noncompact Riemann surface : ''H''/Γ. The most common examples are the curves ''X''(''N'') and ''X0''(''N''), associated with the groups Γ(''N'') and Γ0(''N''), where the explicit classical model for ''X0''(''N'') is the '' ''N''. Then Γ0(''N'') is the larger subgroup consisting of matrices that are upper triangular ''modulo'' ''N''. These curves have a direct interpretation as Moduli Space s for Elliptic Curve s, with 'markings'. Thus ''X''(''N'') is the moduli space for elliptic curves with a given basis for the ''N''-torsion, whereas ''X0''(''N'') is the moduli space for elliptic curves with a cyclic subgroup of order ''N''. These curves have been studied in great detail, and in particular, it is known that ''X0''(''N'') can be defined over Q. The equations defining modular curves are the best-known examples of Modular Equation s. The ''best models'' can be very different from those taken directly from Elliptic Function theory. Hecke Operator s may be studied geometrically, as Correspondence s connecting pairs of modular curves. RELATION WITH THE MONSTER Modular curves of . The traditional name for such a generator, which is unique up to a Möbius Transformation and can be appropriately normalized, is a Hauptmodul ('''main''' or '''principal modular function'''). First several coefficients of ''q''-expansions of these functions were computed already in 19th century, but it came as a shock that the same large integers show up as dimensions of representations of the largest sporadic simple group Monster. The relation runs very deep and as demonstrated by Richard Borcherds , it also involves Generalized Kac-Moody Lie Algebra s. Work in this area underlined the importance of Modular ''functions'' that are meromorphic and can have poles at the cusps, as opposed to Modular ''forms'' , that are holomorphic everywhere, including the cusps, and had been the main objects of study for the better part of the 20th century. COMPACTIFICATION The quotient H is not Compact : for example, the path ''τ''=''yi'', ''y''>1 in the upper half-plane will project onto a path on the modular curve that does not have a limit. Compactification can be easily effected by considering the action of Γ on the extension ''H'' ∪ Q ∪ ∞i of the upper half-plane by "points at infinity". The equivalence classes of the points at infinity under the Γ action are called the '''cusps''' of Γ. They are finite in number, have neighborhoods analytically isomorphic to a complex disk, and thus the quotient :(''H'' ∪ Q ∪ ∞i )/Γ is a compact Riemann surface and complete algebraic curve, called a compactified modular curve. Remark: quotients that ''are'' compact do occur for Fuchsian Group s Γ other than subgroups of the modular group; a class of them constructed from Quaternion Algebra s is also of interest in number theory. SEE ALSO
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