8
votes
Accepted
How to compute Coefficients in Chudnovsky's Formula?
Let $\tau$ be any CM point. By basic theorems of complex multiplication, if you choose a suitable period
$\omega(\tau)$, $E_4(\tau)/\omega(\tau)^4$, $E_6(\tau)/\omega(\tau)^6$,
and $\sqrt{D}E_2^*(\tau)...
- 13.1k
7
votes
Accepted
Eisenstein $E_2$ at imaginary quadratic arguments
Exactly the same Chowla--Selberg formula is valid, but you must apply it
to the modified (non-holomorphic) $$E_2^*(\tau)=E_2(\tau)-3/(\pi\Im(\tau))$$
In other words, $E_2^*(\tau)/\eta^4(\tau)$ is an ...
- 13.1k
7
votes
Accepted
Complex Multiplication and algebraic integers
I have seen this statement about $E_2^*$ tossed around off-handedly by experts a number of times, but never seen a complete proof referenced.
The tools to prove it are (mostly) in Masser's "Elliptic ...
- 1,021
6
votes
Accepted
Why does this quasi-modular function have integral values?
Algebraicity is proven in appendix one of [1]. The function considered there is
$$ \psi(\tau) = \frac{3E_4(\tau)}{2E_6(\tau)} (E_2(\tau) - \frac{3}{\pi \rm{Im} \tau}) = \frac{3}{2} s_2(\tau).$$
The ...
- 1,376
5
votes
Complex Multiplication and algebraic integers
The statement about $E_2^*(\tau)$ is Proposition 5.10.6 in Cohen-Strömberg, Modular forms: a classical approach.
- 20.8k
3
votes
Eisenstein $E_2$ at imaginary quadratic arguments
Let $r$ be a positive rational number and let $\tau=(1+i\sqrt{r}) /2$ so that $\exp(2\pi i\tau) =-e^{-\pi\sqrt{r}} =-q$ and then your $E_2(\tau)$ is nothing but $P(-q) $ with $$P(q) =1-24\sum_{n\geq 1}...
- 2,249
2
votes
Modular equations for quasimodular forms
I just stumbled across this old question. I don’t know if you are still interested in it, but here's some thoughts.
If $F(z)$ is any modular form for $\Gamma_0(N)$ of weight $k$, then $F(z)$ has a ...
- 1,021
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