Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 137 / 4 / b

Table of contents
- Solution b

Solution

For

δ \in Γ (1)

, right multiplication by

δ

permutes

Γ_{\infty} \Γ (1)

. The automorphy factor identity

j (γ δ, τ) = j (γ, δ τ) j (δ, τ)

therefore gives

E_{k, s} (δ τ) = j (δ, τ)^{k} E_{k, s} (τ) .

The modular form

f

obeys the same weight-

k

transformation law, while

Im (δ τ)^{k} = \frac{Im ( τ ) ^{k}}{∣ j ( δ , τ ) ∣ ^{2 k}} .

Consequently

f (δ τ) \overline{E_{k, s} (δ τ)} Im (δ τ)^{k} = j (δ, τ)^{k} \overline{j (δ, τ)}^{k} ∣ j (δ, τ) ∣^{- 2 k} f (τ) \overline{E_{k, s} (τ)} Im (τ)^{k} = f (τ) \overline{E_{k, s} (τ)} Im (τ)^{k} .

Thus the product is invariant under the weight-zero action of

Γ (1)

, as described by the invariant product with a weight-k real-analytic Eisenstein series.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version