Computing in Jacobians of projective curves over finite fields

We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that many desirable operations can be done efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite morphisms, and hence Picard and Albanese maps on Jacobians; generating uniformly random divisors and points on Jacobians; computing Frobenius maps and Kummer maps; and finding a basis for the $l$-torsion of the Picard group, where $l$ is a prime number different from the characteristic of the base field.Comment: 42 page

A criterion to rule out torsion groups for elliptic curves over number fields

We present a criterion for proving that certain groups of the form $\mathbb Z/m\mathbb Z\oplus\mathbb Z/n\mathbb Z$ do not occur as the torsion subgroup of any elliptic curve over suitable (families of) number fields. We apply this criterion to eliminate certain groups as torsion groups of elliptic curves over cubic and quartic fields. We also use this criterion to give the list of all torsion groups of elliptic curves occurring over a specific cubic field and over a specific quartic field.Comment: 13 page