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

David T. -B. G. Lilienfeldt

TL;DR

The paper studies the Darmon–Rotger cycle on the triple product X_0(p)^3 over the quadratic field K ramified at p, proving the cycle Xi = Δ_+ − Δ_- is null-homologous and analyzing its symmetry. By constructing and analyzing twisted triple product L-functions L(F,χ,s) with χ the Legendre symbol at p, it shows the global root number W(F,χ) = −1, implying L(F,χ,s) has odd-order vanishing at the center s = 2. Under Beilinson–Bloch–Kato conjectures, this supports the expectation that the Darmon–Rotger cycle could be non-torsion, aligning with the Gross–Zagier philosophy. The work also develops a detailed, representation-theoretic framework via Weil–Deligne representations and local epsilon factors to establish these root-number phenomena, and outlines natural generalizations to composite levels and higher weights alongside connections to p-adic families and generalized Gross–Kudla–Schoen cycles.

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 $χ$ is the Legendre symbol at $p$. We prove that it is null-homologous and describe actions of various groups on it. For any three normalised cuspidal eigenforms $f_1, f_2, f_3$ of weight $2$ and level $Γ_0(p)$, we prove that the global root number of the twisted triple product $L$-function $L(f_1\otimes f_2\otimes f_3\otimes χ, s)$ 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.

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

TL;DR

The paper studies the Darmon–Rotger cycle on the triple product X_0(p)^3 over the quadratic field K ramified at p, proving the cycle Xi = Δ_+ − Δ_- is null-homologous and analyzing its symmetry. By constructing and analyzing twisted triple product L-functions L(F,χ,s) with χ the Legendre symbol at p, it shows the global root number W(F,χ) = −1, implying L(F,χ,s) has odd-order vanishing at the center s = 2. Under Beilinson–Bloch–Kato conjectures, this supports the expectation that the Darmon–Rotger cycle could be non-torsion, aligning with the Gross–Zagier philosophy. The work also develops a detailed, representation-theoretic framework via Weil–Deligne representations and local epsilon factors to establish these root-number phenomena, and outlines natural generalizations to composite levels and higher weights alongside connections to p-adic families and generalized Gross–Kudla–Schoen cycles.

Abstract

We consider an algebraic cycle on the triple product of the prime level modular curve with origins in work of Darmon and Rotger. It is defined over the quadratic extension of ramified only at whose associated quadratic character is the Legendre symbol at . We prove that it is null-homologous and describe actions of various groups on it. For any three normalised cuspidal eigenforms of weight and level , we prove that the global root number of the twisted triple product -function is . 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.
Paper Structure (21 sections, 14 theorems, 96 equations)

This paper contains 21 sections, 14 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. The Darmon--Rotger cycle
  3. Root numbers
  4. The Beilinson--Bloch--Kato conjectures
  5. Composite levels and higher weights
  6. Conventions
  7. Outline
  8. Triple product cycles
  9. Modular curves
  10. Cycles on $X_1(p)^3$
  11. Action of the group ${\rm SL}_2(\mathbb{F}_p)$
  12. Action of the diamond operators
  13. Action of the Galois group ${\rm Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$
  14. Cycles on $X_0(p)^3$
  15. Field of definition
  16. ...and 6 more sections

Key Result

Theorem 1.1

The Darmon--Rotger cycle is null-homologous and its rational equivalence class gives rise to an element $\Xi:=\Delta_+-\Delta_-$ in the Chow group $\mathop{\mathrm{CH}}\nolimits^2(X_0(p)^3_{K})_0^{\tau=-1}.$ The element $\Xi$ is fixed by the action of the symmetric group $S_3$ if $p\equiv 1\pmod 4$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.6
  • ...and 26 more