Timeline for Complex Multiplication and algebraic integers
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 7, 2019 at 21:33 | comment | added | François Brunault | I should add that the purported strategy in my above comment doesn't work as stated since the modular forms which appear do not always have integral coefficients. I think that further knowledge of the theory of CM is needed | |
| Oct 4, 2018 at 11:23 | comment | added | François Brunault | @L.Miller A precision: it is not hard to show as an exercise that the algebra of modular forms with integral coefficients is $\mathbb{Z}[E_4,E_6,\Delta]$ (hint: starting with $f \in M_k(\mathbb{Z})$, substract a suitable monomial in $E_4$ and $E_6$, and then divide by $\Delta$ to reduce the weight). So you just need to check that the modular forms which appear are in $\mathbb{Z}[[q]]$ which shouldnot be too hard. Another reference is Zagier's article in the 1-2-3 of modular forms. | |
| Oct 4, 2018 at 10:52 | comment | added | François Brunault | @L.Miller In principle, you can follow the proof of Prop 5.10.6 to get the minimal polynomial $\sqrt{D} E_2^*(\tau)$. Its coefficients are universal homogeneous polynomials in $E_4,E_6$ so it is a matter of checking whether these universal polynomials have integral coefficients (note that the writing as $P(E_4,E_6)$ is not unique, so you may need to change it). | |
| Oct 4, 2018 at 6:26 | comment | added | L. Milla | Thank you! Unfortunately the proof of 5.10.6 begins with "we only prove algebraicity, not the integrality". How can one prove the integrality? | |
| Oct 4, 2018 at 6:02 | vote | accept | L. Milla | ||
| Mar 6, 2019 at 7:36 | |||||
| Oct 4, 2018 at 0:30 | history | answered | François Brunault | CC BY-SA 4.0 |