From Inverse Picard to Inverse-Mordell Weil
- Series
- Algebra Seminar
- Time
- Thursday, March 31, 2011 - 16:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Pete Clark – University of Georgia
Which commutative groups can occur as the ideal class group (or
"Picard group") of some Dedekind domain? A number theorist naturally
thinks of the case of integer rings of number fields, in which the
class group must be finite and the question of which finite groups
occur is one of the deepest in algebraic number theory. An algebraic
geometer naturally thinks of affine algebraic curves, and in
particular, that the Picard group of the standard affine ring of an
elliptic curve E over C is isomorphic to the group of rational points
E(C), an uncountably infinite (Lie) group. An arithmetic geometer
will be more interested in Mordell-Weil groups, i.e., E(k) when k is a
number field -- again, this is one of the most notorious problems in
the field. But she will at least be open to the consideration of E(k)
as k varies over all fields.
In 1966, L.E. Claborn (a commutative algebraist) solved the "Inverse
Picard Problem": up to isomorphism, every
commutative group is the Picard group of some Dedekind domain. In the
1970's, Michael Rosen (an arithmetic geometer) used elliptic curves to
show that any countable commutative group can serve as the class group
of a Dedekind domain. In 2008 I learned about Rosen's work and showed
the following theorem: for every commutative group G there is a field
k, an elliptic curve E/k and a Dedekind domain R which is an overring
of the standard affine ring k[E] of E -- i.e., a domain in between
k[E] and its fraction field k(E) -- with ideal class group isomorphic
to G. But being an arithmetic geometer, I cannot help but ask about
what happens if one is not allowed to pass to an overring: which
commutative groups are of the form E(k) for some field k and some
elliptic curve E/k? ("Inverse Mordell-Weil Problem")
In this talk I will give my solution to the "Inverse Picard Problem"
using elliptic curves and give a conjectural answer to the "Inverse
Mordell-Weil Problem". Even more than that, I can (and will, time
permitting) sketch a proof of my conjecture, but the proof will
necessarily gloss over a plausible technicality about Mordell-Weil
groups of "arithmetically generic" elliptic curves -- i.e., I do not
in fact know how to do it. But the technicality will, I think, be of
interest to some of the audience members, and of course I am (not so)
secretly hoping that someone there will be able to help me overcome
it.