Explicit computations of Serre's obstruction for genus 3 curves and application to optimal curves

Details Explicit computations of Serre's obstruction for genus 3 curves and application to optimal curves Explicit computations of Serre's obstruction for genus 3 curves and application to optimal curves

2009-01-19

arxiv.org/abs/0901.2920v2

Mathematics

Number Theory

24 pages ; added : an explicit model, remarks on the hyperelliptic and decomposable reduction, references

Scientific paper

Let k be a field of characteristic different from 2. There can be an obstruction for an indecomposable principally polarized abelian threefold (A,a) over k to be a Jacobian over k. It can be computed in terms of the rationality of the square root of the value of a certain Siegel modular form. We show how to do this explicitly for principally polarized abelian threefolds which are the third power of an elliptic curve with complex multiplication. We use our numeric results to prove or refute the existence of some optimal curves of genus 3.

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