It is well known that there are plenty possible applications of ‘fast arithmetic’ (that is, 1. having an algorithm at hand that actually computes in…, and 2. the running time of that algorithm is known and not too bad) in the degree zero Picard group (the ‘Jacobian’) of an integral, smooth and projective curve $C$ with small genus (for instance elliptic curves and hyperelliptic curves of genus 2) over a field $k$. For instance, there are cryptosystems whose safety relies on the discrete logarithm problem in the Jacobian of elliptic curves and thus it is crucial to know how fast the arithmetic in that group can be carried out.

I wonder whether there are applications of ‘fast arithmetic’ in the degree zero Picard group of more general projective curves $C$ over $k$. Especially, I am curious about the case when $C$ is at least one of the following: *reducible*, *singular* or has *large genus*.

For instance, in the paper of Michael Stoll and Peter Bruin the authors use the Mordell-Weil sieve to prove the non-existence of rational points on smooth projective curves $C$ over $mathbb{Q}$ of genus $g geq 2$. They use the embedding of the rational points on $C$ into the Mordell-Weil group (degree zero Picard group) and local data of reductions at primes $p$ of $mathbb{Q}$. And there is a speed up in their computation if they do not restrict themselves to primes $p$ of good reduction, but also consider $p$ with bad reduction. In that case the authors provide a variant of the Cantor algorithm to compute in the Jacobian in the case that the reduction has genus $2$. Here a generalization to higher genus may be useful to also gather information from bad primes whose corresponding reduction has large(er) genus.

I am interested if there are further applications of ‘fast arithmetic’ in the Jacobian/degree zero Picard group in the same manner as above or even in a complete different style (for instance, it might be valuable to know a lower bound for the running time of such an algorithm to provide statements about how hard a problem related to the Jacobian is).

Many thanks in advance for your help!