Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM. For a good prime $p$, define $\theta_{E}(p)$ by $$\cos\theta_{E}(p)=\frac{p+1-N_{p}(E)}{2\sqrt{p}}\quad (0\leq \theta_{E}(p)\leq \pi).$$ I wonder if the function $\theta_{E}$ is injective?


No. If $E_p$ is a supersingular elliptic curve and $p>3$ then trace of Frobenius on $E_p$ is zero, so $\theta_E(p)=\pi/2$.

By a result of Elkies any elliptic curve over $\mathbb{Q}$ has supersingular reduction in infinitely many primes, so for infinitely many $p$ this function takes the same value.

  • 13
    $\begingroup$ On the other hand, this is the only answer. If $(p+1-N)/\sqrt{p} = (q+1-M)/\sqrt{q}$ with $p+1-N$ and $q+1-M$ nonzero, then $\sqrt{p/q}$ is rational, which it isn't. So the only solutions are when $p+1-N=q+1-M=0$, the supersingular case. $\endgroup$ – David E Speyer Oct 10 '16 at 13:26

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.