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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What if there aren’t enough eta products?<br />
Answer: There are. Philosophically, Ligozat works in different<br />
weights, <strong>and</strong> we can apply the θ operator. Weight 0 to weight 2<br />
(functions to differentials) is especially straight<strong>for</strong>ward.<br />
Theorem 4: (Shimura) Let f be a <strong>for</strong>m of even weight k. Let<br />
ν = f (z)(dz) k/2 . Then<br />
(∑ )<br />
Div(f ) = Div(ν) + (k/2) · (1 − e<br />
−1<br />
i<br />
)P i + cusps ,<br />
where {P i } are the elliptic points of order e i .