For elliptic curves with large 3-torsion Galois image, the Selmer-rank distribution over S3-cubic extensions with a fixed quadratic resolvent is a parity mixture of a universal Markov-chain law, giving a 31.95% lower bound for small rank growth.
Stark points and p -adic iterated integrals attached to modular forms of weight one
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Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents
For elliptic curves with large 3-torsion Galois image, the Selmer-rank distribution over S3-cubic extensions with a fixed quadratic resolvent is a parity mixture of a universal Markov-chain law, giving a 31.95% lower bound for small rank growth.