I am looking for a list about the known or unknown complexity of various number theoretic /algebraic problems. For example,
- GCD in $NC^1$ is open,
- factoring in $P$ is open,
- computing sheaf cohomology is $\#P$-hard,
- Arora and Barak state that a variant of factoring is $NP$-complete (though this is not clear based on the discussion at An NP-complete variant of factoring.),
- Barbulescu et al's breakthrough work on discrete logarithms.
Does anyone know a list of (major) discoveries since these lists were published?
What are some problems of a number theoretic/algebraic flavor whose complexity classes are possibly already known (since the above lists were published), unknown but conjectured, or unknown and not conjectured?
Some avenues of problems could be interpolation problems (univariate or multivariate, over various fields), Chinese remainder theorem, complexity of point counting over curves, etc.