Defining equations for hyperelliptic Jacobians in a neighbourhood of the identity

2011-04-01T09:14:48Z

Let $X$ be a hyperelliptic curve of genus $g \ge 2$ over a field $k$ (of characteristic not 2, 3 or 5, if you like, but could be positive in general). Let $J$ be the Jacobian of $X$, thought of as $\operatorname{Pic}^2(X)$ with elements expressed as divisors in Mumford form (recall that the identity element is written $[1, 0]$ in this case). To keep things simple, assume $X$ has an odd degree model, so it has a unique point $\infty \in X$ at infinity which is a rational Weierstrass point and so we don't have any technical problems with the Mumford representation. Let $\iota:J\rightarrow \mathbb{P}^n$ be an embedding of $J$ and denote by $\mathcal{J}$ the image $\iota(J)$. By smoothness, the maximal ideal $\mathfrak{m}$ of the identity element $0_{\mathcal{J}} = \iota([1,0])$ of $\mathcal{J}$ is generated by $g$ local parameters $t_1, \ldots, t_g$ .

I am interested in calculating (explicitly, for a given curve $X$) the $t_i$ as well as formal expansions of functions $f \in \mathcal{O}_{\mathcal{J},0_{\mathcal{J}}}$ (i.e. the image of such functions under the canonical map $\mathcal{O}_{\mathcal{J},0_{\mathcal{J}}} \to \widehat{\mathcal{O}}_{\mathcal{J},0_{\mathcal{J}}}$ to the completion at $0_{\mathcal{J}}$ ).

In the case of $g = 2$, Grant [1] (under the assumptions above) and Flynn [2] (under more relaxed assumptions) computed the embedding $\iota$, the local parameters at $0_{\mathcal{J}}$ , and defining equations for $\mathcal{J}$ as a smooth projective variety in $\mathbb{P}^8$ and $\mathbb{P}^{15}$ respectively. With a rational Weierstrass point, $\iota(\mathcal{J})$ is given by 13 defining equations in $\mathbb{P}^8$ (see [1, Corollory 2.14]); in the general case one needs 72 defining equations in $\mathbb{P}^{15}$ (see [2, Theorem 1.2]). Evidently this process is somewhat unwieldy and difficult to generalise (in general, I think one needs to take something like $\mathbb{P}^{4^g - 1}$ as the ambient space for the embedding of $J$ to be smooth).

My question, then, is

Suppose that, instead of requiring that $\iota \colon J \to \mathbb{P}^d$ be an embedding, we simply ask that $\iota$ be a rational map. Can we then explicitly describe an $\iota$ such that (i) $d$ is "small" (at worst polynomial, but preferably linear, in $g$), (ii) the number of defining equations for $\mathcal{J} = \iota(J)$ is "small" (same definition of "small" as for (i)), and (iii) $0_{\mathcal{J}} = \iota([1,0])$ is non-singular?

An example of a result which is very close to what I'm looking for was given by Mumford [3, p3.20] where he shows that there is an embedding $i\colon U \to \mathbb{A}^4$ , where $U = J - \Theta$ (where $\Theta$ is the theta divisor---the image of the curve in the Jacobian), defined by sending $[x^2 + a_1 x + a_2, y - (b_1 x + b_2)]$ to $(a_1, a_2, b_1, b_2)$ . Of course, this $U$ doesn't include the identity $[1, 0]$ , so I can't use it to solve the problem above. My attempts to adapt it have thus far been fruitless.

Any ideas would be most welcome.

[1] Grant, D., "Formal groups in genus two." J. Reine Angew. Math. 411 (1990), 96–121.

[2] Flynn, E. V., "The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field." Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 3, 425–441.

[3] Mumford, D., Tata lectures on theta, II. Progress in Mathematics, 43. Birkhäuser Boston, Inc., Boston, MA, 1984.

Answer by Hamish Ivey-Law for references for abelian schemes

2011-03-20T22:19:43Z

The equations defining the Jacobian of a curve as a projective variety become very complicated as soon as the genus of the curve is bigger than 1. In the case of genus 2 curves, say $\mathcal{C}:y^2 = f(x)$, Grant [1] gives an explicit embedding in $\mathbb{P}^8$ and the defining equations when $\deg(f) = 5$ and Flynn [2] gives an explicit embedding in $\mathbb{P}^{15}$, the 72 (!) defining equations of the projective variety, and the biquadratic forms defining the addition law for when $\deg(f) = 6$ (see also Cassels and Flynn's book [3] for an "updated" version of Flynn's work in the early 90s among other things). For this reason, most computations with Jacobians use the Mumford representation of points in $\operatorname{Sym}^2(\mathcal{C})$ together with Cantor's algorithm for the addition law.

[1] Grant, D. Formal groups in genus two. J. Reine Angew. Math. 411 (1990), 96–121.

[2] Flynn, E. V. The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field. Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 3, 425–441.

[3] Cassels, J. W. S.; Flynn, E. V. Prolegomena to a middlebrow arithmetic of curves of genus 2. London Mathematical Society Lecture Note Series, 230. Cambridge University Press, Cambridge, 1996. xiv+219 pp. ISBN: 0-521-48370-0

Comment by Hamish Ivey-Law

2011-04-01T15:11:58Z

@mdeland: You're right, I meant to write "embedding". This now fixed.

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Defining equations for hyperelliptic Jacobians in a neighbourhood of the identity

Answer by Hamish Ivey-Law for references for abelian schemes

Comment by Hamish Ivey-Law