Tag Archives: random matrix

Maples on Cohen-Lenstra for matrices with iid entries

Emmanuel Kowalski pointed me to a very interesting recent paper by Kenneth Maples, a grad student at UCLA working under Terry Tao.  One heuristic justification for the Cohen-Lenstra conjectures, due to Friedman and Washington, relies on the remarkable fact that if M is a random nxn matrix in M_n(Z_p), the distribution of coker(M) among finite abelian p-groups approaches a limit as n goes to infinity; so it makes sense to talk about “the cokernel of a large random matrix” without specifying the size.  (There’s a fuller discussion of Friedman-Washington in this old post.)

Maples shows that the requirement that M is random — that is, that the entries of M are independently drawn from Z_p with additive Haar measure —  is much stronger than necessary.  In fact, he shows that when the entries of M are drawn independently from any distribution on Z_p satisfying a mild non-degeneracy condition, the distribution of coker(M) converges to the so-called Cohen-Lenstra distribution, as in Friedman-Washington.  That’s pretty cool!  I don’t know any arithmetic circumstance that would naturally produce exotic distributions of this kind, but the result helps bolster one’s psychological sense that the Cohen-Lenstra distribution provides the only sensible notion of “cokernel of random matrix,” in some robust sense.

Universality of random matrix laws is a very active and fast-moving topic, but Maples’ result is the first universality result for p-adic matrices that I know of.  More generally, I think there’s a lot to be gained by understanding how well the richly developed theory of random large real and complex matrices carries over to the p-adic case.

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How many points does the average curve have?

Felipe Voloch complained that I didn’t list Ruby’s BBQ in my last post as one of the charms of visiting UT. I’ll make it up to him by observing that one of the charms of visiting UT is talking math with Felipe! He asked me an interesting question, about which we had different intuitions — I’ll present the question here and those readers who have an opinion are encouraged to voice it. (Math below the fold to avoid shocking the modesty of non-mathy readers.)

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