On the rank of elliptic curves

112 citations recorded by the source corpus.

The paper presents an approach that may lead to a counterexample to the Birch and Swinnerton-Dyer conjecture.

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Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves (2017, methodology): “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].”
Some structure theorems for algebraic groups (2015, background): “(Such a pair (E,L) exists unless k is algebraic over a finite field, as follows from [55]; see also [59]).”
Class Field Towers (2010, background): “General questions about connections between elliptic curves and class groups are discussed in Billing [311], Brumer & Kramer [339], Buell [340], Bölling [351], Satgé [354], Quer [355], Frey [358], Aoki [364], Soleng [366] abd Sato [373].”
Elliptic Surfaces (2009, background): “In fact, Shafarevich’s original idea of proof generalises to almost all cases.”