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1ÒΓ(1)ÓÖØÙÐÐÑÓÙÐÖÖÓÙÔÓÑ<strong>×</strong>ÖÓѵºΓ(N)<strong>×</strong>¸ÒظÒÓÖÑÐ<strong>×</strong>ÙÖÓÙÔÓ ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

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Γ(N)ºÌÒÜÓΓ(N)ÒΓ(1)<strong>×</strong>ØÒÙÑÖÓÕÙÚÐÒÐ<strong>×</strong><strong>×</strong><strong>×</strong>ÓÑØÖ<strong>×</strong><br />

Γ(1)¸<strong>×</strong>Ò<strong>×</strong>ÐÝÚÖÝ<strong>×</strong>ÒØØA−1 BA ∈ Γ(1)ÓÖÒÝØÛÓÑØÖ<strong>×</strong>A ÒB∈<br />

ÈË<strong>×</strong>ØØ<strong>×</strong>ÖØÓÐÐÓÛÒ<strong>×</strong>ÙÖÓÙÔ<strong>×</strong>ÓΓ(1)<br />

´º¾µ<br />

}<br />

ÛÖÑÒ<strong>×</strong>ÒÝÐÑÒغÌÒÙÑÖN<strong>×</strong>ÐÐØÐÚÐÓΓºÏÒÒÑÓÙÐÖ ´º¿¼µ<br />

(ÑÓN) ,<br />

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(ÑÓN) ,<br />

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ÚØÓÖ<strong>×</strong>ÔVºÌÖÖ<strong>×</strong>ÚÖÐÛÝ<strong>×</strong>ÓÒÒÐØØÒÚØÓÖ<strong>×</strong>Ô¸ÛÚÓÒ ØØ<strong>×</strong><strong>×</strong><strong>×</strong>ØØÓÙÒÖ<strong>×</strong>ØÒÐÓÛº <strong>×</strong>ÙØØV/L<strong>×</strong>ÓÑÔغËÑÐÖÐݸÓÒÒÒÐØØÒÓÑÔÐÜÚØÓÖ<strong>×</strong>Ôº<br />

2<strong>×</strong>ÙØØ ÐØØÒÖÐÚØÓÖ<strong>×</strong>ÔVÓÒØÑÒ<strong>×</strong>ÓÒ<strong>×</strong><strong>×</strong>ÖØ<strong>×</strong>ÙÖÓÙÔ¸L¸ÓV ËÔÐÐݸÓÒ<strong>×</strong>ÖCºÚÒØÛÓÒÓÒ¹ÚÒ<strong>×</strong>ÒÓÑÔÐÜÒÙÑÖ<strong>×</strong>ω 1Òω )<strong>×</strong><br />

ω 1 /ω 2 /∈ R¸ÛÒ<strong>×</strong><strong>×</strong>ÓØÐØظL(ω , ω 2 )¸ØÓω Ï<strong>×</strong><strong>×</strong>ÙÑÁÑ(ω 2 /ω 1 ) > 0º{ω 1 , ω 2 }<strong>×</strong>Ø<strong>×</strong><strong>×</strong>ÓLºÄØMÒÓØØ<strong>×</strong>ØÓÔÖ<strong>×</strong>(ω ÓÐÑÒØ<strong>×</strong>ÓC∗¸ÒÐØLØ<strong>×</strong>ØÓÐÐÐØØ<strong>×</strong>ÓCºÌÑÒÓÐC/L(ω 1 , ω 2<br />

½¾<br />

{ ( ) (<br />

a b<br />

Γ 1 (N) =<br />

∈ SL(2, Z) :<br />

c d<br />

{ ( ) ( ) ( )<br />

a b<br />

a b ∗ ∗<br />

Γ 0 (N) =<br />

∈ SL(2, Z) : ≡<br />

c d<br />

c d 0 ∗<br />

{ ( ) ( ) ( )<br />

Γ 0 a b<br />

a b ∗ 0<br />

(N) =<br />

∈ SL(2, Z) : ≡<br />

c d<br />

c d ∗ ∗<br />

∈ Γ(1)<br />

´º¾µ<br />

) ( )<br />

a b 1 ∗<br />

≡ (ÑÓN)}<br />

c d 0 1<br />

1 1Òω 2ÝL(ω<br />

1<br />

}<br />

}º<br />

1 , ω 2 ) ≡ {Zω 1 + Zω 2<br />

, ω 2 )

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