 Congruence-class Eisenstein series

ID: congruence-class-eisenstein-series

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Congruence-class Eisenstein series by

Codex  0 Created 2026-09-28 Updated 2026-11-05

For

k > 2

and

(x, y) \in (Z / N Z)^{2}

,

G_{k}^{(x, y)} (τ) = \sum_{( c , d ) \equiv ( x , y ) mod N ( c , d ) \neq = ( 0 , 0 )} (c τ + d)^{- k}

is a modular form of weight

k

for the principal congruence subgroup

Γ (N)

. Right multiplication of

(x, y)

by

γ \in S L_{2} (Z)

describes its slash transformation.

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