Eta Products and Models for Modular Curves
Eta Products and Models for Modular Curves
Eta Products and Models for Modular Curves
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Moreover, the values of f 24 at the d = 18, 6, 3, <strong>and</strong> 2 cusps are<br />
3 24 , 3 12 , 3 12 , <strong>and</strong> 1 (respectively, up to a root of unity). These<br />
values can be easily verified (in this case) by choosing eta<br />
products which vanish at the other cusps, <strong>and</strong> then comparing<br />
with f .<br />
x = η2 1 η 6η 9<br />
η 2 η 3 η 2 18<br />
(x) = (0) − (∞) x = f − 3<br />
y = η 2η 6 3<br />
η 2 1 η2 6 η3 18<br />
(y) = c 6,1 + c 6,2 − 2(∞) y = f 2 + 3<br />
z = η 1η 8 6 η3 9<br />
η 2 2 η4 3 η6 18<br />
(z) = c 3,1 + c 3,2 − 2(∞) z = f 2 − 3f + 3