Cubic Equations over Function Fields or Rings

I have a question regarding the Weierstrass-type cubic equation

$$Y^2 = 4 X^3 - g_2(z) X - g_3(z),$$

where the functions $$g_2(z)$$ and $$g_3(z)$$ belong to some function field $$\mathcal{F}$$ or ring $$\mathcal{R}$$. For example, $$\mathcal{F}$$ can be the field $$\mathbb{C}(z)$$ of rational functions and $$\mathcal{R}$$ the ring $$\mathbb{C}[z]$$ of polynomials (or, even, $$\mathcal{F}$$ can be the field of meromorphic functions and $$\mathcal{R}$$ the ring of entire functions).

Naturally, one can consider the associated Diophantine-type problem, namely to look for solutions $$X = X(z)$$, $$Y = Y(z)$$ belonging to the field $$\mathcal{F}$$ or to the ring $$\mathcal{R}$$.

Is there any literature on such problems? For example, is there a Mordell-type or a Siegel-type theorem?

• Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves" has a chapter on elliptic surfaces. That is a nice place to start. And yes, there is a lot of literature on that problem.... – Chris Wuthrich Jun 12 at 8:47
• Thanks for the response. Could you suggest some paper(s)? Is there an analog to Mordell's Theorem? – vassilis papanicolaou Jun 13 at 12:48
• Actually, Chapter III of Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves" seems very relevant to my question! Thanks again. – vassilis papanicolaou Jun 13 at 12:59