Selmer ranks under quadratic twists satisfying the Heegner hypothesis

TL;DR

This paper studies how the ranks of Selmer groups of elliptic curves change under quadratic twists that satisfy the Heegner hypothesis, providing explicit formulas and confirming parity conjecture predictions.

Contribution

It derives explicit matrix-based formulas for Selmer rank variations under quadratic twists satisfying the Heegner hypothesis, linking them to the parity conjecture.

Findings

Formulas for Selmer rank changes under quadratic twists

Compatibility with parity conjecture predictions

Explicit matrix descriptions over _2

Abstract

We investigate variations of Selmer ranks under quadratic twists satisfying the Heegner hypothesis. In particular, starting with an elliptic curve $E/\mathbb{Q}$ with partial $2$-torsion and a common relaxed Selmer group, we derive explicit formulae describing the effect of twisting on Selmer ranks in terms of matrices over $\mathbb{F}_{2}$. As an application, we show that these formulae are compatible with predictions made by the parity conjecture.