- Series
- Other Talks
- Time
- Wednesday, November 2, 2011 - 5:15pm for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- David Brown – Department of Mathematics and Computer Science, Emory University
- Organizer
- Douglas Ulmer

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.