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Modular Forms, Eisenstein series, show it transforms with modular of weight 2

  1. May 6, 2017 #1
    1. The problem statement, all variables and given/known data

    I need to show that

    e2.jpg

    transforms with modular of weight ##2## for ##SL_2(Z)##

    We have the theorem that it is sufficient to check the generators S and T st.jpg

    We have that E_2 is (whilst holomorphic) fails to transform with modular weight ##2## as it has this extra term when checking for ##S##:

    here.png

    where transforming with weight ##2## means (for ##S##):

    ##f(S.\tau)=\tau^{2}f(\tau)##


    Therefore we expect a cancellation from the ##Im (\tau)## term

    We have
    sol.png




    MY QUESTION
    - I follow this solution and this is fine
    - I guess I am doing something pretty stupid, but we have the following formula
    im tau.png
    for ##G=SL_2(R)## and since ##Z \in R## we have ##SL_2(Z) \in SL_2(R)##, so the above also holds for ##SL_2(Z) ## and so this gives:

    ##Im(S.\tau)=\frac{Im(\tau)}{\tau^2}##


    - so ##1/Im(\tau)## transforms with modular weight ##2## itself, and so is not cancelling
    - I perhaps thought there may have been an issue that ##\infty## is not taken into account, but it also says the action on ##G## extends to ##\infty## , as I've said I follow the above solution, but have no idea what is wrong with using this formula.

    Many thanks in advance

    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. May 11, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
  4. May 16, 2017 #3
    ta bot bruv'
     
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