Bad GRE with an otherwise good record - is it possible my application will not even be considered? I will try my best to remain anonymous. I have studied my bachelor and master's degree in a university outside the US. The math department is very strong and my adviser is well known - he has several papers with over 200 citations. I have good grades with many advanced courses, in fact I believe I have already studied beyond what is required in some PhD programs. My MSc thesis features some new results known to experts but not otherwise. I believe I will have good letters of recommendation.

I am trying to apply to the top 10 math departments. I believe that my record is overall strong but my GRE scores are extremely bad, in fact much below average. I did my math GRE under very bad circumstances, both physically and personally. Is it possible that my application won't even be considered by the committee? I am worried I won't pass the initial filter and that nobody would even look at my letters of recommendation. Is this reasonable?
In my experience with graduate admissions in mathematics departments, all complete applications are looked at by departmental faculty (and mildly incomplete ones may be looked at as well, albeit more briefly).  So I think it is likely that your letters will be read.

However, you say that your GRE scores are "much below average".  I will assume that your description is somewhat factually based and thus that they are below the 35th percentile.  In my department -- about the 50th best -- we would like to see higher scores than that, but we can take chances on some applicants for whom that is the only weak point.  Admission to the top 10 math departments in the US is much more competitive than this: they can fill their classes with applicants who are extremely successful according to all the standard metrics.  If they deviate from that then they will do so only for someone who is truly exceptional.  

You say that your "thesis features some new results known to experts but not otherwise".  I don't mean to be harsh, but this sounds quite equivocal: to a research mathematician "new" implies that it is not known to experts.  I suppose you would have mentioned if you had published your results, so I will say that it is often difficult to publish such results in a reputable journal.  Anyway "truly exceptional" here means that you have results that were not just unknown to everyone but are actually surprising / exciting to the experts, and that they are eager to build on your work and learn more about your ideas.  It sounds like what you've done is write an ordinarily strong master's thesis, and in my opinion this is not going to offset your poor GRE scores.

My advice is to apply to a lot of departments.  Every individual committee gets to decide how much to value GRE scores; I have certainly met individual faculty members who do not place much stock in them, so while it is unlikely that an entire admissions committee would feel that way, it must happen sometimes.  If you don't get into a good enough department, you should consider deferring for a year, retaking the GRE under better conditions and really nailing it.  If you really do feel that you can do much better on it, then waiting one year and then going to a top department is probably a better career move in the long run.  However, if you wait a year, take the GRE again and don't do better, it will be disappointing, so you should make an effort to figure out whether you will really do better, which may involve taking practice tests but also some soul-searching and personal honesty.

Good luck.  And to everyone else who is reading this: yes, for math PhD students, the GRE is a very serious business.  Please take it seriously and plan to do well, rather than trying to figure out how to paint over this weakness in your application.