Completion of the proof of theorem 3.3 in Dale Husemoller: Elliptic Curves

Question

I want to read the proof of the following theorem:

This is from p.35. But it is not complete there. There is written that:

Can someone tell me where I can find the rest of the proof?

Any other sources are also welcome :)

Thanks in advance

Answer 1

Claim: Let $E: y^2=x^3+D$ and $p>3$ be a prime. Then, there is no point of order $p$ in $E(\mathbb{Q})$.

Here are some hints. Let $p>3$ be a prime as in the statement of the claim:

• If $q$ is a prime such that $q\equiv 2 \bmod 3$, and $q$ does not divide $6D$, then $E(\mathbb{F}_q)=q+1$.

• A prime $q$ that does not divide $6D$ is a prime of good reduction for $E$. Thus, $E(\mathbb{Q})[m]$ embeds into $E(\mathbb{F}_q)$ when $\gcd(m,q)=1$.

• In particular, if $E(\mathbb{Q})[p]$ is non-trivial, then $q+1$ is divisible by $p$, for all primes $q\equiv 2\bmod 3$ and $q>6D$. In other words, every prime $q\equiv 2 \bmod 3$ with $q>6D$ satisfies $q\equiv -1 \bmod p$ (contradiction!).

Answer 2

As Álvaro Lozano-Robledo already showed, to prove that there is no $p$-torsion it is enough to find a prime $q \not\mid 6D$ such that $q \equiv 2 \bmod 3$ and $q \not\equiv -1 \bmod p$. The punchline he left unstated is that the existence of such $q$ is guaranteed by Dirichlet's theorem on primes in arithmetic progressions. Choose $m_0 \not\equiv 0, -1 \bmod p$ (say $m_0=1$). By the Chinese Remainder Theorem, there exists $m \equiv m_0 \bmod p$ such that $m \equiv 2 \bmod 3$. Then $m$ is coprime to $3p$, so by Dirichlet there are infinitely many primes $q \equiv m \bmod 3p$. Since there are only finitely many factors of $6D$, it follows that there are primes $q \equiv 2 \bmod 3$ that are neither factors of $6D$ nor congruent to $-1 \bmod p$, and we're done.