I'm interested in the complexity of solving linear equations modulo k, for arbitrary k (and with a special interest in prime powers), specifically:
Problem. For a given system of $m$ linear equations in $n$ unknowns modulo $k$, do there exist any solutions?
In the abstract to their paper Structure and importance of logspace-MOD classes on the classes ModkL, Buntrock, Damm, Hertrampf, and Meinel claim that they "demonstrate their significance by proving that all standard problems of linear algebra over the finite rings $\mathbb Z/k\mathbb Z$ are complete for these classes". On closer inspection, the story is more complicated. For instance, Buntrock et al. show (by a proof-sketch in an earlier and freely accessible draft found by Kaveh, thanks!) that solving systems of linear equations is instead in the complementary class coModkL, for k prime. This class is not known to be equal to ModkL for k composite, but never mind that — what I'm concerned about is the fact that they don't make any remarks about whether solving systems of linear equations mod k is even contained in coModkL for k composite!
Question: Is solving systems of linear equations modulo k contained in coModkL for all positive k?
If you can solve systems of equations modulo a higher power q of a prime p, you can solve them modulo p as well; so solving systems of equations modulo q is coModpL-hard. If you could show that this problem is in ModqL, you would end up showing ModkL = coModkL for all k. That's likely to be difficult to prove. But is it in coModkL?