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Let $X$ be a curve of genus two over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian of $X$.

Can we write down an explicit equation for the abelian surface $J$?

I know $X$ can be given by the equation $y^2 =f(x)$ with $f(x)\in k[x]$ of degree $5$ or $6$.

(Note that I'm actually asking for an equation for a surface birational to $J$.)

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    $\begingroup$ Flynn, The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field, Math. Proc. Cambridge Philos. Soc. 107 (1990), 425–441. Did google or mathscinet fail to produce this reference? $\endgroup$
    – Felipe Voloch
    Commented Dec 4, 2012 at 20:16
  • $\begingroup$ There are also paper(s) of David Grant, I'll let you look them up. If I remember correctly, both Flynn and Grant not only give equations for J, they also give equations for the group law. $\endgroup$
    – Joe Silverman
    Commented Dec 4, 2012 at 20:30

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In Mumford's ``Tata Lectures on Theta II" (Progress in Math. 43, 1984) there are explicit equations for a certain open affine (dense) subset $Z$ of the jacobian $Jac(C)$ for any hyperelliptic curve $C$; the jacobian is covered by all the translations of $Z$ by points of order $2$. (Recall that every genus 2 curve is hyperelliptic.)

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Every curve of genus $2$ has an equation of the form $Y^{2}Z^{4}=f_{0}X^{6}+f_{1}X^{5}Z+\cdots+f_{6}Z^{6}.$] Flynn (1990) has found the equations of the Jacobian variety of such a curve in characteristic $\neq2,3,5$ --- they form a set $72$ homogeneous equations of degree $2$ in $16$ variables (they take $6$ pages to write out). See also the book Cassels and Flynn 1996;

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