User hamish ivey-law - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:27:24Z http://mathoverflow.net/feeds/user/10080 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60271/defining-equations-for-hyperelliptic-jacobians-in-a-neighbourhood-of-the-identity Defining equations for hyperelliptic Jacobians in a neighbourhood of the identity Hamish Ivey-Law 2011-04-01T09:14:48Z 2011-04-01T15:09:22Z <p>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 <code>$\operatorname{Pic}^2(X)$</code> 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 <code>$\iota:J\rightarrow \mathbb{P}^n$</code> 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 <code>$0_{\mathcal{J}} = \iota([1,0])$</code> of $\mathcal{J}$ is generated by $g$ local parameters <code>$t_1, \ldots, t_g$</code>.</p> <p>I am interested in calculating (explicitly, for a given curve $X$) the <code>$t_i$</code> as well as formal expansions of functions <code>$f \in \mathcal{O}_{\mathcal{J},0_{\mathcal{J}}}$</code> (i.e. the image of such functions under the canonical map <code>$\mathcal{O}_{\mathcal{J},0_{\mathcal{J}}} \to \widehat{\mathcal{O}}_{\mathcal{J},0_{\mathcal{J}}}$</code> to the completion at <code>$0_{\mathcal{J}}$</code>).</p> <p>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 <code>$0_{\mathcal{J}}$</code>, and defining equations for $\mathcal{J}$ as a <em>smooth</em> projective variety in $\mathbb{P}^8$ and $\mathbb{P}^{15}$ respectively. With a rational Weierstrass point, <code>$\iota(\mathcal{J})$</code> is given by 13 defining equations in <code>$\mathbb{P}^8$</code> (see [1, Corollory 2.14]); in the general case one needs 72 defining equations in <code>$\mathbb{P}^{15}$</code> (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 <code>$\mathbb{P}^{4^g - 1}$</code> as the ambient space for the embedding of $J$ to be smooth).</p> <p>My question, then, is</p> <blockquote> <p>Suppose that, instead of requiring that <code>$\iota \colon J \to \mathbb{P}^d$</code> be an embedding, we simply ask that $\iota$ be a rational map. Can we then explicitly describe an <code>$\iota$</code> such that (i) $d$ is "small" (at worst polynomial, but preferably linear, in $g$), (ii) the number of defining equations for <code>$\mathcal{J} = \iota(J)$</code> is "small" (same definition of "small" as for (i)), and (iii) <code>$0_{\mathcal{J}} = \iota([1,0])$</code> is non-singular?</p> </blockquote> <p>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 <code>$i\colon U \to \mathbb{A}^4$</code>, where $U = J - \Theta$ (where $\Theta$ is the theta divisor---the image of the curve in the Jacobian), defined by sending <code>$[x^2 + a_1 x + a_2, y - (b_1 x + b_2)]$</code> to <code>$(a_1, a_2, b_1, b_2)$</code>. Of course, this $U$ doesn't include the identity <code>$[1, 0]$</code>, so I can't use it to solve the problem above. My attempts to adapt it have thus far been fruitless.</p> <p>Any ideas would be most welcome.</p> <hr> <p>[1] Grant, D., "Formal groups in genus two." <em>J. Reine Angew. Math.</em> 411 (1990), 96–121.</p> <p>[2] Flynn, E. V., "The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field." <em>Math. Proc. Cambridge Philos. Soc.</em> 107 (1990), no. 3, 425–441.</p> <p>[3] Mumford, D., Tata lectures on theta, II. Progress in Mathematics, 43. Birkhäuser Boston, Inc., Boston, MA, 1984.</p> http://mathoverflow.net/questions/51955/references-for-abelian-schemes/59004#59004 Answer by Hamish Ivey-Law for references for abelian schemes Hamish Ivey-Law 2011-03-20T22:19:43Z 2011-03-20T22:19:43Z <p>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.</p> <p>[1] Grant, D. <em>Formal groups in genus two</em>. J. Reine Angew. Math. 411 (1990), 96–121.</p> <p>[2] Flynn, E. V. <em>The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field.</em> Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 3, 425–441. </p> <p>[3] Cassels, J. W. S.; Flynn, E. V. <em>Prolegomena to a middlebrow arithmetic of curves of genus 2</em>. London Mathematical Society Lecture Note Series, 230. Cambridge University Press, Cambridge, 1996. xiv+219 pp. ISBN: 0-521-48370-0</p> http://mathoverflow.net/questions/60271/defining-equations-for-hyperelliptic-jacobians-in-a-neighbourhood-of-the-identity Comment by Hamish Ivey-Law Hamish Ivey-Law 2011-04-01T15:11:58Z 2011-04-01T15:11:58Z @mdeland: You're right, I meant to write &quot;embedding&quot;. This now fixed.