Consider the quadratic forms $$ Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2 $$ over the rational function field $\mathbb{F}_p(t)$, where $p > 2$ is a prime such that $t^2 + 1$ is irreducible over $\mathbb{F}_p$, i.e., $p \equiv 3 \ (\mathrm{mod} \ 4)$.

These forms are isomorphic over $\mathbb{F}_p(t)$, because their Hilbert symbols $(-1,t^2+1)_v$, $(-1,1)_v$ are equal for all valuations $v$ of $\mathbb{F}_p(t)$.

How explicitly write a projective transformation (over $\mathbb{F}_p(t)$) between $Q_1$ and $Q_2$? Thank you in advance.

  • 2
    $\begingroup$ Both conics have a rational point ($(t,1,1)$ for $Q_{1}$, $(1,0,1)$ for $Q_{2}$). Projection from this point gives an explicit isomorphism $u_{i}:Q_{i}\rightarrow \mathbb{P}^1$. Just take $u_{2}^{-1}\circ u_{1}$. $\endgroup$ – abx Jan 23 '19 at 5:14

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