Upper bounds for torsion in class groups
(Un exposé de A. Venkatesh au séminaire de
théorie des nombres de Chevaleret le 5 juin 2006)
Résumé :
I will discuss the problem of giving upper bounds on the $l$-torsion part
of the class group of a number field.
One believes that this is "quite small" relative to the size of the
entire class group; such results
are quite useful (e.g. to bound ranks of elliptic curves by descent)
but seem to be quite hard to come by. For instance, for $Q(\sqrt{-D})$
the size of the entire class group is about $D^{1/2}$; one believes that
the size of the 3-torsion part is $<< D^{\epsilon}$, and the best known
bound is $D^{1/3}$.
I will discuss some of the existing work (by L. Pierce, Heath-Brown,
Helfgott/Venkatesh) and then discuss recent work with Jordan Ellenberg,
which gives some new results (such as the $D^{1/3}$ mentioned above, as
well as certain results for fields of higher degree).