I hope this is a good question.

Recently I worked with genus two curves $H$ that have multiplication by $[\zeta_5]\in \text{Aut}(H)$, that is, multiplication by $e^{2\pi i/5}$. This automorphism is naturally extended to the Jacobian of $H$. Since $\zeta_5 + \zeta_5^{-1}=\tfrac{-1+\sqrt{5}}{2}$ and the Endomorphism ring of the Jacobian of $H$ is isomorphic to $\mathbb{Z}[\zeta_5]$ there is a $\sqrt{5}$ isogeny, which I computed explicitly to work over Finite fields using the curve $y^2 = x^5 + c$.

The question is, I was reading Paul Gaudry's paper on how to do scalar multiplication on the Kummer Surface. In his paper he uses Rosenhain form of hyperelliptic curves (and my curve is not in that form obviously).

https://hal.inria.fr/inria-00000625v2/document

I am a little lost but, can someone point me out if I can do this scalar multiplication by $\sqrt{5}$ using the Kummer surface ? More precisely, if $D_1,D_2\in\mathcal{J}$ are divisors in the Jacobian of a hyperelliptic curve of genus 2. Is there a way to obtain analog formulas like the biquadratic forms for $D_1+D_2$ or $2D_1$ in the Kummer surface but for $\sqrt{5}D_1$ in the Kummer?

Thanks