List of number theoretic or algebraic problems in various complexity classes
Algebraic geometry


Noether's Normalization Lemma (NNL) for explicit varieties is currently only known to be in $\mathsf{EXPSPACE}$ (like general NNL), but is conjectured to be in $\mathsf{P}$ (and is in $\mathsf{P}$ assuming that PIT can be black-box derandomized). Update 4/18/18: It was recently shown that for the variety $\overline{\mathsf{VP}}$ it is in $\mathsf{PSPACE}$ over the rationals (Forbes &amp; Shpilka) and then over arbitrary fields (Guo, Saxena, &amp; Sinhababu).
Testing whether a given set of polynomials has an algebraic dependence. This problem was recently shown to be in $\mathsf{AM} \cap \mathsf{coAM}$ by  Guo, Saxena, &amp; Sinhababu (improving the previous upper bound of $\mathsf{NP}^{\mathsf{\# P}}$ due to Mittmann, Saxena, &amp; Scheblechner, also on the arXiv).
There are several (arXiv) new algorithms for computing topological invariants of complex varieties (with various restrictions like smoothness, etc.). I believe for most of these the optimal upper bound is still open.
Hilbert's Nullstellensatz (HN): given integer polynomials, decide if they have a common complex solution, is in $\mathsf{AM}$ assuming GRH (Koiran). It is unknown if it is in $\mathsf{NP}$.
Algorithms for resolution of singularities of algebraic varieties in characteristic zero. The current best time upper bound, due to Bierstone, Grigoriev, Milman, and Włodarczyk is $\mathcal{E}^{d+3}$ where $d$ is the dimension of the variety and $\mathcal{E}^{\bullet}$ is the Grzegorczyk hierarchy of primitive recursive functions. There aren't particularly good (any?) lower bounds on this problem, but for a seemingly much simpler problems related lower bounds are known, namely: There are ideals in $n$ variables generated in degree at most $n$ that require $\mathcal{E}^{n+1}$ such generators. So the current upper bound for resolution of singularities may not be far from the truth, but little is really known.


Isomorphism problems


Many problems on permutation groups - such as coset intersection, permutation group isomorphism, etc. - are in $\mathsf{NP} \cap \mathsf{coAM}$, but it is unknown if they are in $\mathsf{NP} \cap \mathsf{coNP}$, and it is suspected that they are not in $\mathsf{P}$. Graph Isomorphism reduces to most of those problems, so a better upper bound on them implies a better upper bound on GI. 
In particular, for permutation group isomorphism, the current best upper bound is $2^{O(n)}|G|$, and it is open if it can be done in $2^{O(n)}$ time (depending only on the degree of the permutation group and not on its order), let alone quasi-poly time like GI and coset intersection.
Group Isomorphism where groups are given by multiplication tables is known to be in $\mathsf{TIME}(n^{O(\log n)})$, but is suspected to be in $\mathsf{P}$. Known to be in $\mathsf{P}$ for several classes of groups (update 4/18/18: and a couple (arXiv) more (arXiv)), but not in general.


Other


Update 4/18/18: Tensor rank over any field $\mathbb{F}$ is $\exists \mathbb{F}$-complete (Schaefer &amp; Stefankovic). Is tensor rank over $\mathbb{Q}$ in $\mathsf{NP}$? It is known to be $\mathsf{NP}$-hard (Håstad), and over finite fields it is in $\mathsf{NP}$.
More generally, many problems on tensors over $\mathbb{Q}$ are $\mathsf{NP}$-hard but not known to be in $\mathsf{NP}$ (Hillar and Lim, also on the arXiv).


It would (somewhat sadly) seem that the Adleman-McCurley survey, despite being 21 years old, is pretty up to date in terms of number-theoretic problems, with the exception of the fact that we now know that $\mathsf{PRIMES}$ is in $\mathsf{P}$...