For elliptic curves with vanishing 2-adic mu-invariant and small lambda-invariant, the authors show that many quadratic twists have Selmer corank 1, conditionally yielding rank 1 curves when the Tate-Shafarevich group is finite.
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Iwasawa theory and ranks of elliptic curves in quadratic twist families
For elliptic curves with vanishing 2-adic mu-invariant and small lambda-invariant, the authors show that many quadratic twists have Selmer corank 1, conditionally yielding rank 1 curves when the Tate-Shafarevich group is finite.