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Twisted triple product root numbers and a cycle of Darmon-Rotger
Number Theory · Submitted 8 Oct 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For any three normalised cuspidal eigenforms $f_1, f_2, f_3$ of weight $2$ and level $\Gamma_0(p)$, we prove that the global root number of the twisted triple product $L$-function $L(f_1\otimes f_2\otimes f_3\otimes \chi, s)$ is $-1$.
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The abstract, sentence by sentence
We consider an algebraic cycle on the triple product of the prime level modular curve $X_0(p)$ with origins in work of Darmon and Rotger. It is defined over the quadratic extension of $\mathbb{Q}$ ramified only at $p$ whose associated quadratic character $\chi$ is the Legendre symbol at $p$. We prove that it is null-homologous and describe actions of various groups on it. For any three normalised cuspidal eigenforms $f_1, f_2, f_3$ of weight $2$ and level $\Gamma_0(p)$, we prove that the global root number of the twisted triple product $L$-function $L(f_1\otimes f_2\otimes f_3\otimes \chi, s)$ 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider an algebraic cycle on the triple product of the prime level modular curve $X_0(p)$ with origins in work of Darmon and Rotger. | contribution 0.64 | 0.56 | 0.04 | asserted 0.95 |
| s1 | It is defined over the quadratic extension of $\mathbb{Q}$ ramified only at $p$ whose associated quadratic character $\chi$ is the Legendre symbol at $p$. | background 0.53 | 0.33 | 0.03 | asserted 1.00 |
| s2 | We prove that it is null-homologous and describe actions of various groups on it. | result 0.90 | 0.95 | 0.16 | asserted 1.00 |
| s3 | For any three normalised cuspidal eigenforms $f_1, f_2, f_3$ of weight $2$ and level $\Gamma_0(p)$, we prove that the global root number of the twisted triple product $L$-function $L(f_1\otimes f_2\otimes f_3\otimes \chi, s)$ is $-1$. | result 0.99 | 0.95 | 0.57 | asserted 1.00 |
| s4 | 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. | interpretation 0.96 | 0.21 | 0.14 | hedged possibility 1.00 |
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The paper
Original abstract
We consider an algebraic cycle on the triple product of the prime level modular curve $X_0(p)$ with origins in work of Darmon and Rotger. It is defined over the quadratic extension of $\mathbb{Q}$ ramified only at $p$ whose associated quadratic character $\chi$ is the Legendre symbol at $p$. We prove that it is null-homologous and describe actions of various groups on it. For any three normalised cuspidal eigenforms $f_1, f_2, f_3$ of weight $2$ and level $\Gamma_0(p)$, we prove that the global root number of the twisted triple product $L$-function $L(f_1\otimes f_2\otimes f_3\otimes \chi, s)$ 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.
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