references for abelian schemes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T09:49:45Z http://mathoverflow.net/feeds/question/51955 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51955/references-for-abelian-schemes references for abelian schemes unknown 2011-01-13T13:45:08Z 2011-03-20T22:19:43Z <p>Hi, I have a very basic question. I am looking for references explaining how to construct explicitily a Jacobian starting from a curve or examples of projective equations for an abelian scheme. I know a little bit the theory in general so I need examples to fix it, at least in the cases which are not too complicate( or when it is possible). Thank you </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>