Thursday, March 31, 2011 - 16:00
1 hour (actually 50 minutes)
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.