Let $A/L$ be an elliptic curve, with complex multiplication by a quadratic imaginary field $K$.

A theorem by Deuring ([13, paragraph 4], Theorem 12 on page 182 of Elliptic Functions by Serge Lang) states that if $\mathfrak{P}$ is a prime of $L$ lying over $p$ at which $A$ has good reduction, then $A$ is supersingular at $\mathfrak{P}$ precisely when there is exactly one prime of $K$ lying over $p$. This means we can "read off" the ordinary or supersingular reduction of $A/L$ by looking at the extension $K/\mathbb{Q}$. (See also here: reduction of CM elliptic curves)

However, Silverman's book "Advanced topics in the arithmetic of elliptic curves" says the following (Exercise II.2.30(c) (proof of theorem II.10.5(b)), alternatively, see here https://wstein.org/129-05/final_papers/Corina_Patrascu.pdf, proof of Theorem 13):

Suppose $K \not\subseteq L$. Let $\mathfrak{P}$ be a prime of $L$ at which $A$ has good reduction. Then the reduction of $A$ at $\mathfrak{P}$ is supersingular if and only if $\mathfrak{P}$ is inert or ramifies in $LK/L$. Hence, this would imply we can read off ordinary or supersingular reduction from the extension $LK/L$.

Although $LK/L$ and $K/\mathbb{Q}$ have somewhat similar splitting behaviour (if a prime splits completely in $L/\mathbb{Q}$, then the behaviour in $LK/L$ and $K/\mathbb{Q}$ is the same), differences can occur when $p$ is (somewhat) inert in $L/\mathbb{Q}$.

My question would then be: where did I make a mistake? Did I misinterpret or misread one of these two theorems?


1 Answer 1


It seems like Silverman's exercise is wrong: see this errata.


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