Georgia Institute of Technology College of Sciences

Knowledge of the distribution of class groups is elusive -- it is not even known if there are infinitely many number fields with trivial class group. Cohen and Lenstra noticed a strange pattern -- experimentally, the group \mathbb{Z}/(9) appears more often than \mathbb{Z{/(3) x \mathbb{Z}/(3) as the 3-part of the class group of a real quadratic field \Q(\sqrt{d}) - and refined this observation into concise conjectures on the manner in which class groups behave randomly. Their heuristic says roughly that p-parts of class groups behave like random finite abelian p-groups, rather than like random numbers; in particular, when counting one should weight by the size of the automorphism group, which explains why \mathbb{Z}/(3) x \mathbb{Z}/(3) appears much less often than \mathbb{Z}/(9) (in addition to many other experimental observations). While proof of the Cohen-Lenstra conjectures remains inaccessible, the function field analogue -- e.g., distribution of class groups of quadratic extensions of \mathbb{F}_p(t) -- is more tractable. Friedman and Washington modeled the \el$-power part (with \ell \neq p) of such class groups as random matrices and derived heuristics which agree with experiment. Later, Achter refined these heuristics, and many cases have been proved (Achter, Ellenberg and Venkatesh). When $\ell = p$, the $\ell$-power torsion of abelian varieties, and thus the random matrix model, goes haywire. I will explain the correct linear algebraic model -- Dieudone\'e modules. Our main result is an analogue of the Cohen-Lenstra/Friedman-Washington heuristics -- a theorem about the distributions of class numbers of Dieudone\'e modules (and other invariants particular to \ell = p). Finally, I'll present experimental evidence which mostly agrees with our heuristics and explain the connection with rational points on varieties.