| Résume | 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). |
