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I Modular form quick question translation algebra

  1. Dec 22, 2016 #1
    Please see attached.

    I am trying to show that

    ## T_{p} f (\tau + 1) = T_{p} f (\tau ) ##
    ##f(\tau) \in M_k ## and so can be written as a expansion as ##f(\tau)=\sum\limits^{\infty}_{0}a_{n}e^{2 \pi i n \tau } ##
    ##f(\tau + 1) = f(\tau) ## since ##e^{2\pi i n} = 1##
    Similarly ##f(p\tau + p) = f(p\tau) ## for the same reason since ##np \in Z \geq 1 ## so the extra exponential term is ##1## again.

    But I DONT UNDERSTAND how it goes from ##f(\frac{\tau + 1 + j}{p}) = f(\frac{\tau+j}{p}) ## , since it is not guarenteed that ##1/p## is an integer, I mean it only is when ##p=1## so ##e^{2 \pi n i (t+1+j)/p} = e^{ 2 \pi i n (t+j)/p}e^{2 \pi i n (1/p) } ## and ##e^{2 \pi i n (1/p) } ## is equal to ##1## only when ##p=1##. second equality of attached.

    help please. thank you.
     

    Attached Files:

  2. jcsd
  3. Dec 22, 2016 #2

    fresh_42

    Staff: Mentor

    It is simply a substitution of ##j \rightarrow i-1## and calling the ##i## afterwards ##j## again. (A usual business when dealing with sums.)
     
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