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modular form quick question translation property algebra really

Please see attached.

I am trying to show that Tpf(τ+1)=Tpf(τ) T_{p} f (\tau + 1) = T_{p} f (\tau ) f(τ)∈Mkf(\tau) \in M_k and so can be written as a expansion as

f(τ)=/sum/limits0∞ane2πinτf(\tau)=/sum/limits^{\infty}_{0}a_{n}e^{2 \pi i n \tau }

f(τ+1)=f(τ)f(\tau + 1) = f(\tau) since e2πin=1e^{2\pi i n} = 1

Similarly f(pτ+p)=f(pτ)f(p\tau + p) = f(p\tau) for the same reason since np∈Z≥1np \in Z \geq 1 so the extra exponential term is 11 again.

But I DONT UNDERSTAND how it goes from:

f(τ+1+jp)=f(τ+jp)f(\frac{\tau + 1 + j}{p}) = f(\frac{\tau+j}{p}) ,

since it is not guaranteed that 1/p1/p is an integer, I mean it only is when p=1p=1 so
hecke op.png26.8KB
Please clear the mess in your post. But anyways, the last equality holds because the sum is changed from {j=0 to p-1} to {j=1 to p}. If you don't see why this is the same, write out some terms.
Original post by Math12345
Please clear the mess in your post. But anyways, the last equality holds because the sum is changed from {j=0 to p-1} to {j=1 to p}. If you don't see why this is the same, write out some terms.


oh right.

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