Parity over the biquadratic field Q(i,17)\mathbb Q(i,\sqrt{17})

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Let K=Q(−1,17)K=\mathbb Q(\sqrt{-1},\sqrt{17}). Biquadratic-field parity conjecture. Every elliptic curve E/QE/\mathbb Q has even Mordell–Weil rank over KK:

rk⁡E/K≡0(mod2).\operatorname{rk}E/K\equiv0\pmod2.

The proposed statement follows from the parity conjecture because every place of Q\mathbb Q splits into an even number of places in KK, giving global root number +1+1. Its unconditional truth is open with the parity conjecture.

References

Primary source

Tim Dokchitser, “Notes on the Parity Conjecture”, arXiv:1009.5389 (2012).

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