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

Abstract

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.

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

References in corpus (4)

• Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields
• On a conjecture of Serre on abelian threefolds
• Optimal Curves of Genus 3 over Finite Fields with Discriminant -19
• On the Torelli problem and Jacobian Nullwerte in genus three

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• Optimal curves of genus 1,2 and 3
• Large Galois images for Jacobian varieties of genus 3 curves
• Spanning the isogeny class of a power of an ordinary elliptic curve over a finite field. Application to the number of rational points of curves of genus $\le 4$
• Deducing information about curves over finite fields from their Weil polynomials