In the US graduate universities, Ph.D. students have to take courses. Can a student with low GPA do good research? I think doing research is solving problems. If someone cannot solve a homework or an exam, how the person can even solve more complicated problems to publish?
In the US graduate universities, Ph.D. students have to take courses. Can a student with low GPA do good research?
In short, yes. GPA is a measure of your ability to follow instructions, do homework on time, and perform on exams. If the grading scheme has enough of a curve it barely does that. In any case, it's not a meaningful measure of research ability.
In the US graduate program I'm familiar with (in physics), several of the classes you take are actually not that closely related to your own research. Spending too much time on those classes can actively hurt your research output, while being somewhat beneficial to your GPA. Often there are also teaching duties to handle, so time management can be crucial. Sometimes you hear the advice to get good enough grades*, and spend the rest of your time on research. It also seems that potential employers are not too interested in the graduate GPA - the degree itself and your research tend to be more important factors.
*The grades should be enough to remain in good academic standing, and count towards your degree.
I think doing research is solving problems. If someone cannot solve a homework or an exam, how the person can even solve more complicated problems to publish?
To some extent this is true, particularly for the courses relevant to one's research area. Understanding that subject matter and being able to solve problems is clearly worthwhile. Usually that translates to good grades in those classes, but I don't think it's guaranteed. One can have personal issues that semester, suffer from time pressure on a final exam, etc.
On the other hand, research problems are not always just more complicated versions of the problems studied in class. They may require new, creative methods that you wouldn't see in a course. They may require learning specific procedures that just aren't covered in standard courses. Not to mention that being able to pick a good open problems to work on is an art in itself. Certainly, getting good scores in a course does not guarantee one becoming a successful, independent researcher.
In conclusion, there might be some correlation between graduate GPA and research ability for large enough population sizes, but I don't think it holds any predictive power on the individual level.
Sure, it's entirely possible to have a poor GPA and still do great in research!
Maintaining a high GPA and doing great research are two different skill sets. Sure, they have overlapping components - like, as you said, problem solving helps with both - but it's still entirely possible to be weak in one while strong in the other.
Some of the components are even anti-correlated. Examples:
A student who does things exactly the way the teacher wants them to be done is likely to get a good GPA for it, but the student who figures out their own strategies'll tend to do better in the research world.
A student who seeks social validation might work hard on their GPA, but the student who's more interested in the field itself is more likely to excel in research.
A student who thinks in contrived logic should tend to do better on exams, but a student who thinks more generally should do better in research.
Many academics can pull off both a great GPA and great research, but many others will be good at one but not the other.
I would like to add to some of the other good answers already posted here.
Generally speaking in the US, PhD students take classes concurrently with their research with an advisor. As a simple mathematical fact, time spent on classes takes away from time spent on research. Therefore the argument
If someone cannot solve a homework or an exam, how the person can even solve more complicated problems to publish?
ignores the possibility that a student has decided to spend the time on solving a homework problem elsewhere, such as on their research topic. While a good GPA can be a sign of competence, a PhD student with an overly stellar GPA begs the question of why their time was not spent elsewhere.