Pointwise Bound for \ell-torsion of Class Groups

Abstract: The \ell-torsion conjecture states that the size of the \ell-torsion subgroup Cl_K[\ell] of the class group of a number field K is bounded by Disc(K)^{\epsilon}. It follows from a classical result of  Brauer-Siegel, or even earlier result of Minkowski, that the class number |Cl_K| of a number field K is always bounded by Disc(K)^{1/2+\epsilon}, therefore we obtain a trivial bound (Disc}(K)^{1/2+\epsilon} on |Cl_K[\ell]|. We will talk about results on this conjecture, and recent work on breaking the trivial bound for \ell-torsion of class groups in some cases based on the work of Ellenberg-Venkatesh.