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ÔØÖºÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑÝÓÒËÔØÖ ÌÙ<strong>×</strong>¸(f 2 , f 3<br />

+ÒÓØØÙØÙÖÐعÓÒÒØ<strong>×</strong>ÔÒ<br />

2,1ºÓÒ<strong>×</strong>ÖØØѹÐÖÓÒ<br />

, f −2 )ÔÖÓÚ<strong>×</strong><strong>×</strong>ÓÖÅÒÓÛ<strong>×</strong><strong>×</strong>ÔR ´ºµ<br />

V = {x ∈ R 2,1 | (x, x) < 0} ,<br />

ÒR ÁÒØÓÒØÓØØÖÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>¸ØÖÖØÖÔÖÑØÚÐعÐÚØÓÖ<strong>×</strong>¸ºº<br />

2,1ºÄØV )ÛØÑÙÐØÔÐØÝ9ÒØÛÓÔÖÑØÚ<br />

2¸ØËÐÙÔÔÖ¹Ð<strong>×</strong>Ô℄º <strong>×</strong>ÙØØZ∈R<br />

ÒÖØÝØÓÖÑÙÐ<br />

2,1 + iV +ºÌ<strong>×</strong><strong>×</strong>ÕÙÚÐÒØØÓZ 0ºÌÑÒÖÝÐعÐÖÓÓØ<strong>×</strong>Ö<br />

)ºÌ<strong>×</strong>ÖÓÓØ<strong>×</strong>ÖÑÒÖÝ<br />

H<br />

(η, η) = 02f 2¸2f −2Ò(2f −2 − 2f 3 + 2f 2<br />

ÚØÓÖ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÝÒ(η, η) < 0(2f −2 + 2f 2 )Ò(2f −2 − f 3 + 2f 2<br />

<strong>×</strong>ÒØÖÒÓÖÑ<strong>×</strong>ÒÓØÔÓ<strong>×</strong>ØÚÒظºº(η, η) ≤<br />

ÆØÚÚÐÙÓÑÙÐØÔÐØÝÑÔÐ<strong>×</strong>ØØØÖÓÓØ<strong>×</strong>ÖÑÓÒºÓÖÒ<strong>×</strong>ØÒ¸ÓÒ<strong>×</strong> ´ºµ<br />

−27ºÌÙ<strong>×</strong>¸<strong>×</strong>ÙÖÓÓØ<strong>×</strong>ÖÖÑÓÒÒÒÛÚ<strong>×</strong>ÙÔÖÐÖºÌ<br />

IIºÄØÙ<strong>×</strong>ÐÓÓØØÏÝÐÖÓÙÔÓ<br />

.<br />

η(τ) 3<br />

º¿º½ 1º<br />

m(2η 0 ) =<br />

ÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÐÓÒØÓØ<strong>×</strong>ÔM II ∩ R + M<br />

ØÃÅÄ<strong>×</strong>ÙÔÖÐÖG )¸<strong>×</strong>ØÖÓÙÔÒÖØÝØØÖÐÑÒØÖÝÖØÓÒ<strong>×</strong>¸<br />

)¸ÛÓ<strong>×</strong>ÖÑÑØÖÜ<strong>×</strong>ÚÒÝØÑØÖÜ ÌÏÝÐÖÓÙÔW(A 1,II )<br />

ÚÒØØÖÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>(δ 1 , δ 2 , δ 3<br />

A 1,II¸ØÏÝÐÖÓÙÔ¸W(A<br />

±1ºÇÒ<strong>×</strong>℄´<strong>×</strong>Ð<strong>×</strong>Ó½¼¸½℄µ Z)<strong>×</strong>ÚÒÝØÒØÖÐ )Ò<br />

(w 1 , w 2 , w 3<br />

´ºµ<br />

)¸ÛØÖ<strong>×</strong>ÔØØÓØØÖÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÌÏÝÐÖÓÙÔW(A ÛÖØØÒ<strong>×</strong>ÒÓÖÑÐ<strong>×</strong>ÙÖÓÙÔÓPGL(2, Z)ºÊÐÐØØPGL(2, ÑØÖ<strong>×</strong>( a<br />

c d )ÛØad bc =<br />

⊗Q<strong>×</strong>ÙØØÐÐØÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong> ÛÖS 3<strong>×</strong>ØÖÓÙÔÓÔÖÑÙØØÓÒ<strong>×</strong>ÓØØÖÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÐ<strong>×</strong>Ó¸ØÐØØM II<br />

<strong>×</strong>ÐØØÏÝÐÚØÓÖÛ<strong>×</strong>ÒÐÑÒØρ∈M II<br />

½¼<br />

1 − ∑ t∈N<br />

Z = z 3 f 2 + z 2 f 3 + z 1 f −2 ,<br />

∈<br />

m(tη 0 )q t = ∏ k∈N(1 − q k ) 9 =<br />

√<br />

f<br />

(10)<br />

(τ)<br />

1,II<br />

1,II<br />

b<br />

−<br />

PGL(2, Z) = W(A 1,II ) ⋊ S 3 ,

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