# The existence of infinitely many supersingular primes for every elliptic curve over Q

Elkies proved The existence of infinitely many supersingular primes for every elliptic curve over Q. I read his paper, but found the supersingular primes he constructed are all 3(mod 4) type. So, how about the others?

It's the auxiliary prime $l$ that must be $3 \bmod 4$; the supersingular prime $p$ is not guaranteed to be congruent to $3 \bmod 4$, and indeed the residue of $p \bmod 4$ is unpredictable (unless the curve has CM by an order in ${\bf Q}(i)$). Several examples on page 565 have $p \equiv 1 \bmod 4$. For example, the curve $X_0(11)$: $y^2+y=x^3-x^2$ has $j = -2^{12}/11$ and bad reduction only at $11$, so we can take $l=7$ because the Legendre symbol $(11/7)$ is $+1$. The polynomial $P_l(X)$ is then $P_7(X) = X + 15^3$, and we find $P_7(j) = 33029/11$, with $(33029/7) = -1$. Hence some prime factor of $33029$ is supersingular. It so happens that $33029$ is itself prime, and congruent to $1 \bmod 4$.
Having ordinary or supersingular reduction at $p$ for CM elliptic curves depends on the splitting type of $p$ in $\mathrm{End}(E) \otimes \mathbf{Q}$, see reduction of CM elliptic curves
• So a curve like $y^2=x^3-x$, for instance, will only have supersingular reduction at $p\equiv 3$ mod $4$. Nov 23, 2013 at 17:21
• @Timo: For CM curves, one knows what's going on. Elkies paper is really about non-CM curves. I hadn't realized that the infinitely many ss primes he constructs are all 3 (mod 4). For a non-CM curve over $\mathbb{Q}$, I see no reason why there shouldn't be infinitely many ss primes in any arithmetic progression (containing infinitely many primes, of course). But possibly there are some local constraints. Maybe Noam can chime in. Nov 23, 2013 at 17:26