There is the following result of Deuring that goes as follows:

Let $E/L$ be an elliptic curve defined over a number field $L$ with complex multiplication by an order $\mathcal{O}$ in an imaginary quadratic field $K$, $\mathfrak{p}$ a prime in $\overline{\mathbb{Q}}$ over a rational prime $p$, and $\tilde{E}$ the reduction of $E$ modulo $\mathfrak{p}$. Then $p$ is a supersingular prime for $E$ if and only if $p$ is ramified or inert in $K$.

I have two questions:

• What is an intuitive way to see that this result is true?
• What is the quickest way to prove this result?

Many thanks in advance. Perhaps this question is a bit silly because it is not very hard assuming the main results of complex multiplication...

There is the following result of Deuring that goes as follows:

Let $E/L$ be an elliptic curve defined over a number field $L$ with complex multiplication by an order $\mathcal{O}$ in an imaginary quadratic field $K$, $\mathfrak{p}$ a prime in $\overline{\mathbb{Q}}$ over a rational prime $p$, and $\tilde{E}$ the reduction of $E$ modulo $\mathfrak{p}$. Then $p$ is a supersingular prime for $E$ if and only if $p$ is ramified or inert in $K$.

I have two questions:

• What is an intuitive way to see that this result is true?
• What is the quickest way to prove this result?

Many thanks in advance. Perhaps this question is a bit silly because it (is not very hard assuming) assumes the main results of complex multiplication...

There is the following result of Deuring that goes as follows:

Let $E/L$ be an elliptic curve defined over a number field $L$ with complex multiplication by an order $\mathcal{O}$ in an imaginary quadratic field $K$, $\mathfrak{p}$ a prime in $\overline{\mathbb{Q}}$ over a rational prime $p$, and $\tilde{E}$ the reduction of $E$ modulo $\mathfrak{p}$. Then $p$ is a supersingular prime for $E$ if and only if $p$ is ramified or inert in $K$.

I have two questions:

• What is an intuitive way to see that this result is true?
• What is the quickest way to prove this result?

Many thanks in advance.

There is the following result of Deuring that goes as follows:

Let $E/L$ be an elliptic curve defined over a number field $L$ with complex multiplication by an order $\mathcal{O}$ in an imaginary quadratic field $K$, $\mathfrak{p}$ a prime in $\overline{\mathbb{Q}}$ over a rational prime $p$, and $\tilde{E}$ the reduction of $E$ modulo $\mathfrak{p}$. Then $p$ is a supersingular prime for $E$ if and only if $p$ is ramified or inert in $K$.

I have two questions:

• What is an intuitive way to see that this result is true?
• What is the quickest way to prove this result?

Many thanks in advance. Perhaps this question is a bit silly because it is not very hard assuming the main results of complex multiplication...