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<p>We consider an algebraic cycle on the triple product of the prime level modular curve <i>X</i><sub>0</sub>(<i>p</i>) with origins in work of Darmon and Rotger. It is defined over the quadratic extension of ℚ ramified only at <i>p</i> whose associated quadratic character <i>χ</i> is the Legendre symbol at <i>p</i>. We prove that it is null-homologous and describe actions of various groups on it. For any three normalized cuspidal eigenforms <i>f</i><sub>1</sub>, <i>f</i><sub>2</sub>, <i>f</i><sub>3</sub> of weight 2 and level Γ<sub>0</sub>)(<i>p</i>), we prove that the global root number of the twisted triple product <i>L</i>-function <i>L</i>(<i>f</i><sub>1</sub> ⊗ <i>f</i><sub>2</sub> ⊗ <i>f</i><sub>3</sub> ⊗ <i>χ, s</i>) is −1. Assuming conjectures of Beilinson and Bloch, and guided by the Gross–Zagier philosophy, this suggests that the Darmon–Rotger cycle could be non-torsion, although we do not currently have a proof of this.</p>

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Twisted triple product root numbers and a cycle of Darmon–Rotger

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  • 发表日期：

    2026年6月21日

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  • David T.-B. G. Lilienfeldt

摘要

We consider an algebraic cycle on the triple product of the prime level modular curve X0(p) with origins in work of Darmon and Rotger. It is defined over the quadratic extension of ℚ ramified only at p whose associated quadratic character χ is the Legendre symbol at p. We prove that it is null-homologous and describe actions of various groups on it. For any three normalized cuspidal eigenforms f1, f2, f3 of weight 2 and level Γ0)(p), we prove that the global root number of the twisted triple product L-function L(f1 ⊗ f2 ⊗ f3 ⊗ χ, s) is −1. Assuming conjectures of Beilinson and Bloch, and guided by the Gross–Zagier philosophy, this suggests that the Darmon–Rotger cycle could be non-torsion, although we do not currently have a proof of this.