Frobenius distributions of low dimensional abelian varieties over finite fields

Given a $g$-dimensional abelian variety $A$ over a finite field $\mathbf{F}_q$, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most $g$. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre--Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre--Frobenius groups that occur for $g \le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $g>3$.Comment: Comments welcom

Optimal curves of genus 1,2 and 3

In this survey, we discuss the problem of the maximum number of points of curves of genus 1,2 and 3 over finite fieldsComment: 18 pages. To appear in "Publications Mathematiques de Besancon(PMB)

The characteristic polynomials of abelian varieties of dimensions 3 over finite fields

We describe the set of characteristic polynomials of abelian varieties of dimension 3 over finite fields

Constructing hidden order groups using genus three Jacobians

Groups of hidden order have gained a surging interest in recent years due to applications to cryptographic commitments, verifiable delay functions and zero knowledge proofs. Recently, Dobson and Galbraith ([DG20]) proposed Jacobians of genus three hyperelliptic curves as a suitable candidate for such a group. While this looks like a promising idea, certain Jacobians are less secure than others and hence, the curve has to be chosen with caution. In this short note, we explore the types of Jacobians that would be suitable for this purpose

Genus 3 curves with many involutions and application to maximal curves in characteristic 2

Let k=F_q be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C_2\times C_2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an Artin-Schreier cover of the three elliptic curves obtained as the quotient of C by the nontrivial involutions of H, and the Jacobian of C is k-isogenous to the product of these three elliptic curves. In this paper we exhibit explicit models for genus 3 curves with many involutions, and we compute explicit equations for the elliptic quotients. We then characterize when a triple (E_1,E_2,E_3) of elliptic curves admits an Artin-Schreier cover by a genus 3 curve, and we apply this result to the construction of maximal curves. As a consequence, when q is nonsquare and m=\lfloor 2 sqrt(q) \rfloor = 1,5,7 mod 8, we obtain that N_q(3)=1+q+3m. We also show that this occurs for an infinite number of values of q nonsquare.Comment: 18 page