The Brauer-relation parity conjecture for Jacobians

Let XX be a curve defined over a number field KK, and let GG be a finite group of KK-automorphisms of XX. Assume that Ω1(JacX)\Omega^{1}(\operatorname{Jac}_X) is self-dual as a GG-representation. Let Θ\Theta be a Brauer relation in GG, let pp be a prime, and let w(XτΘ,p/Kv)w(X^{\tau_{\Theta,p}}/K_v) and ΛΘ(X/Kv)\Lambda_\Theta(X/K_v) denote the associated local root number and regulator-constant term. The Brauer-relation parity conjecture.

v place of Kw(XτΘ,p/Kv)(1)ordpΛΘ(X/Kv)=1.\prod_{v\textup{ place of }K} w(X^{\tau_{\Theta,p}}/K_v) (-1)^{\operatorname{ord}_p \Lambda_\Theta(X/K_v)} = 1.

This is the local-term comparison needed to obtain the parity conjecture for the relevant self-dual representation components of the Jacobian. The supplied text does not state that it has been proved or disproved.

Sources & referencesView supporting material

Primary source

Vladimir Dokchitser, Holly Green, Alexandros Konstantinou and Adam Morgan, “Parity of ranks of Jacobians of curves”, arXiv:2211.06357 (2024).

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