The elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ is interesting as the associated modular form (this is over $\bf Q$) is $$ F=q\prod_{n>0}(1-q^n)^2(1-q^{11n})^2 $$ Clearly this exhibits a very nice $\eta$-product.

Is anyone aware of other elliptic curves over $\bf Q$ which have a simple minimal equation and whose associated modular form is a nice $\eta$-product or even a nice $\eta$-quotient?

  • $\begingroup$ Every weight $2$ eta product which is a modular form is the L-function of an elliptic curve. Such a formula in term of $\eta$ is useful for the BSD conjecture. Not sure if that it doesn't vanish on the upper half-plane is helpful. $\endgroup$ – reuns Sep 6 '19 at 19:45

Yes, there are a few elliptic curves whos modular forms exhibit such nice factorizations in terms of the $eta$-function. One example that I (re)found a while ago is


which has the associated modular form


You can view more of these here. And while trying to find that PDF again, I also found this had been asked previously on math overflow.

| cite | improve this answer | |
  • $\begingroup$ Ah thank you for the link, it is exactly what I'm looking for. $\endgroup$ – Kevin Sep 6 '19 at 14:54
  • $\begingroup$ Do you mean the MO-link or the first link with the paper by Ono? $\endgroup$ – Dietrich Burde Sep 6 '19 at 14:54
  • $\begingroup$ @DietrichBurde Actually both, the MO question is weirdly similar to mine, even the conductor is the same. $\endgroup$ – Kevin Sep 6 '19 at 14:55

For several values of $\lambda\in \Bbb Q\setminus \{0,1\}$ the elliptic curves

$$E_{\lambda}\colon y^2=x(x-1)(x-\lambda)$$ correspond to modular forms which are linear combinations of eta-quotients, see here. For more details see the MO question, also linked by $dx dy dz$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.