Let M be the Monster simple group. Let o(g), g∈M, be the order of g. Define Γ0(N)={(abcd)∈SL(2,Z)/c≡0 (mod N)}, Γ(N)={(abcd)∈SL(2,Z)/(abcd)≡(1001) (mod N)} and let Λ(N) be the normalizer of Γ0(N) in SL(2,R). We shall call a discrete subgroup Δ of SL(2,R) a congruence group if it contains Γ(N) in Δ is finite and Δ acts on the extended upper half plane. H*=H∪Q∪{i∞} by fractional linear transformations. The quotient Δ. H* has the structure of a compact Riemann surface and will be denoted by X(Δ). Conway and Norton [CN] conjectured relationships between certain congruence groups and M known as the moonshine conjectures. These have been proved by Borcherds:
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机译：令M为Monster简单组。令o（g），g∈M为g的阶数。定义Γ0（N）= {（abcd）∈SL（2，Z）/c≡0（mod N）}，Γ（N）= {（abcd）∈SL（2，Z）/（abcd）≡（1001 ）（mod N）}，令Λ（N）为SL（2，R）中Γ0（N）的归一化。如果SL（2，R）的离散子群Δ中包含Γ（N）是有限的并且Δ作用在扩展的上半平面上，则我们将其称为同余群。 H * =HQ∪{i∞}通过分数线性变换。商Δ。 H *具有致密的黎曼曲面的结构，将由X（Δ）表示。 Conway和Norton [CN]猜想某些同等群与M之间的关系，即M，称为月光猜想。这些已由Borcherds证明：
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