Jacobians among Abelian threefolds: a formula of Klein and a question of Serre

Details Jacobians among Abelian threefolds: a formula of Klein and a question of Serre Jacobians among Abelian threefolds: a formula of Klein and a question of Serre

2008-02-27

arxiv.org/abs/0802.4017v1

Mathematics

Number Theory

15 pages

Scientific paper

Let k be a field and f be a Siegel modular form of weight h \geq 0 and genus g>1 over k. Using f, we define an invariant of the k-isomorphism class of a principally polarized abelian variety (A,a)/k of dimension g. Moreover when (A,a) is the Jacobian of a smooth plane curve, we show how to associate to f a classical plane invariant. As straightforward consequences of these constructions, when g=3 and k is a subfield of the complex field, we obtain (i) a new proof of a formula of Klein linking the modular form \chi_{18} to the square of the discriminant of plane quartics ; (ii) a proof that one can decide when (A,a) is a Jacobian over k by looking whether the value of \chi_{18} at (A,a) is a square in k. This answers a question of J.-P. Serre. Finally, we study the possible generalizations of this approach for g>3.

Affiliated with

Also associated with

No associations

Rating

Jacobians among Abelian threefolds: a formula of Klein and a question of Serre does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Jacobians among Abelian threefolds: a formula of Klein and a question of Serre, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Jacobians among Abelian threefolds: a formula of Klein and a question of Serre will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-25880