# How explicitly write a projective transformation between the conics over the univariate function field?

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.

• 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 $u_{2}^{-1}\circ u_{1}$. – abx Jan 23 at 5:14