7:25 The formula should be multiplied by m (and so 9:41 the term in blue circle should be T_m/m. Of course this does not affect the proof as T_m/m is still a modular function): From last lecture we have T_m(\sum c_n q_n) = \sum_{k|m} \sum_{j=0^{d-1}} \sum_n c_n q^{nm/k^2} e^{2 pi i nj/d}. Summing over j, we obtain \sum_{k|m} \sum_n k c(nk) q^{nm/k}. Change k -> m/k, this becomes \sum_{k|m} \sum_n k c(nm/k) q^{nk}. Change nk -> n, we get the claimed formula for T_m : \sum_{k|m} \sum_{k|n} m/k c(nm/k^2) q^n
I recall it was said in one of the previous lectures that the constant term of j "should" be 24. So, is there any reason for why we're studying j-744 instead of j-720?
Wow! This may be the only place on the Internet where the definition of hecke operators isn’t immediately followed by wife for a, atkin lehner theory and products of L functions
I love modular forms and always found them interesting. Thank you for your wonderful lecture
Modular forms and number theory in general is just a reminder how fucking spooky 👻 numbers are. I find this shit truly horrifying ...
Great lecture, the interpretation as a Weyl denominator formula was truly unexpected!
I love how around minute 6 he casually brings up the correspondence with the monster group, _which he won the Fields Medal for_.
Thank you for making these videos. I'm learning a lot from watching them.
7:25 The formula should be multiplied by m (and so 9:41 the term in blue circle should be T_m/m. Of course this does not affect the proof as T_m/m is still a modular function):
From last lecture we have T_m(\sum c_n q_n) = \sum_{k|m} \sum_{j=0^{d-1}} \sum_n c_n q^{nm/k^2} e^{2 pi i nj/d}.
Summing over j, we obtain \sum_{k|m} \sum_n k c(nk) q^{nm/k}.
Change k -> m/k, this becomes \sum_{k|m} \sum_n k c(nm/k) q^{nk}.
Change nk -> n, we get the claimed formula for T_m :
\sum_{k|m} \sum_{k|n} m/k c(nm/k^2) q^n
This proof was magical
I recall it was said in one of the previous lectures that the constant term of j "should" be 24. So, is there any reason for why we're studying j-744 instead of j-720?
The only point here is that you want to kill the constant term, so with the "correct" definition you would be looking at j-24 :)
Wow! This may be the only place on the Internet where the definition of hecke operators isn’t immediately followed by wife for a, atkin lehner theory and products of L functions
YEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE