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Modular Forms, Dimension, Valence Formula

  1. Apr 17, 2017 #1
    1. The problem statement, all variables and given/known data

    What is the dimension of ##M_{24}##?

    2. Relevant equations

    attached modweightdim.png
    3. The attempt at a solution


    I am confused what the (mod 12) is referring to- is it referring to the ##[k/12]## where the square brackets denote an equivalent class and the ## k \equiv 2## / ##k \notequiv 2## or just the ##[k/12]##?

    I am confused because I thought ##k \equiv 2## (mod 12) only when ##k=24##, so for the dimension ##M_2## we would need to look at the top definition, however clearly the bottom has been used, which makes me think that the '(mod ##12##)' is only referring to the square brackets?

    In which case for ##M_{24}## I need to look at the top line and conclude ## dim M_{24}=3##, however if (mod 12) is referring to both then I need to look at the bottom line and conclude ##dim M_{24}=2##, however in this case it makes no sense how we have got ##dim M_2=0 ##

    Thanks .
     
  2. jcsd
  3. Apr 17, 2017 #2

    fresh_42

    Staff: Mentor

    ##24 \equiv 0 \operatorname{mod}12##
    ##2 \equiv 2 \operatorname{mod}12##
     
  4. Apr 20, 2017 #3
    So
    ##4 \equiv 4 \operatorname{mod}12##

    So from the definition above ##dim M_{4} =[k/12]=[4/12]##;

    how is ##[4/12]## 1? Isn't this zero too? what do the square brackets denote.

    E.g ##14\equiv 2## (mod 12) so am I using the original ##k## : ##[14/12]## or ##[2/12]##?
     
  5. Apr 20, 2017 #4
    Oh it doesn't matter, [ ] denote equivalent classes, so it's 'the remainder of the division' which is ##2## in both of these cases?

    Can I just test my understanding here- is ##dim M_{28}=[k/12]+1=5##?
     
    Last edited: Apr 20, 2017
  6. Apr 20, 2017 #5
    No I'm lost ##dim M_{12}=2## but I am getting:

    ##12 \equiv 0## mod 12, so I'm looking at ##[k/12]+1##, if these [ ] denote equivalent classes a number divisible by 12 is represented by the element ##0## so I get ##0+1=1##...

    Unless these [ ] square brackets denote taking the integer or something? what do these square brackets mean? thanks.
     
  7. Apr 20, 2017 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    The brackets are the floor function. The greatest integer less than or equal to the quotient.
     
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