Reflection Principles and Bounds for Class Group Torsion

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Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves (2017, background): “This significantly improves on the previously best known bounds of Oǫ(|Disc(E)|1/2+ǫ), cǫ+(1/2+ ǫ) log2(|Disc(E)|), and Oǫ(|Disc(E)|.2007...+ǫ), respectively, as obtained in the works of Brumer and Kramer [4] and Helfgott and Venkatesh [9].”
An effective Chebotarev density theorem for families of number fields, with an application to $$\ell $$-torsion in class groups (2017, background): “The only other known pointwise bounds for prime ` are: n = 2 and ` = 3, where initial progress occurred in [HV06], [Pie05], [Pie06], and [EV07] holds the record C2,3(1/3); C3,3(1/3) due to [EV07]; C4,3(1/2 − δ) due to [EV07] where δ = 1/168 if K is nonD4; Cn,2(0.”
On ℓ-torsion in class groups of number fields (2016, methodology): “This approach has been generalized by Ellenberg and Venkatesh [EV07] to number fields of any degree; we recall the key result in the special case of field extensions of Q: Theorem A (Lemma 2.3 of [EV07]).”
Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points (2011, background): “• The above methods cannot recover some transfer principles, such as the Scholtz reflection principle which relates 3-torsion in the quadratic fields Q( √ D) and Q( √ −3D), or the work of Ellenberg and Venkatesh [5].”