Show that the curve $2Y^2 = X^4-17$ has points in every $\mathbb{Q}_p$

Question

I've been asked to show that the curve $2Y^2 = X^4-17$ has points in every $\mathbb{Q}_p$ - I've managed to show that it is birationally equivalent to the curve $Y^2 = 2X^4 - 34$ (as suggested in the hint) and am told that Hasse's bound is useful but I can't see how to use it - any hints plz?

Answer 1

Hints:

• Prove that the curve has genus 1.

• Find the primes of bad reduction.

• If $p$ is a prime of good reduction, and large enough, then the Hasse-Weil bounds tell you that there is an affine point (not just at infinity) over $\mathbb{F}_p$. Now use Hensel's lemma to lift it up to $\mathbb{Q}_p$.

• If $p$ is a prime of bad reduction, or not large enough, you will have to find $\mathbb{Q}_p$ solutions by hand (with the help of Hensel's lemma, again).