Let K be a number field and let H^1_operatornamemot(K,M)_∞=H^1_operatornamemot(K,M)otimes_FF_∞. The Gillet–Soulé height pairing is the F_∞-bilinear map langle·,·rangle_operatornameGS,∞:H^1_operatornamemot(K,M)_∞× H^1_operatornamemot(K,M)_∞longrightarrow F_∞. Height non-degeneracy conjecture. The pairing langle·,·rangle_operatornameGS,∞ is non-degenerate. The conjecture is motivated by arithmetic analogues of the standard conjectures and by conjectures of Beilinson and Bloch on height pairings.

Non-degeneracy conjecture for the Gillet–Soulé height pairing

About 4 years old · traced to 2022

Let $K$ be a number field and let $H^1_\operatorname{mot}(K,\mathcal{M})_\infty=H^1_\operatorname{mot}(K,\mathcal{M})\otimes_FF_\infty$ . The Gillet–Soulé height pairing is the $F_\infty$ -bilinear map

\langle\cdot,\cdot\rangle_{\operatorname{GS},\infty}:H^1_\operatorname{mot}(K,\mathcal{M})_\infty\times H^1_\operatorname{mot}(K,\mathcal{M})_\infty\longrightarrow F_\infty.

Height non-degeneracy conjecture. The pairing $\langle\cdot,\cdot\rangle_{\operatorname{GS},\infty}$ is non-degenerate. The conjecture is motivated by arithmetic analogues of the standard conjectures and by conjectures of Beilinson and Bloch on height pairings.

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Non-degeneracy conjecture for the Gillet–Soulé height pairing
Let $\mathcal{M}$ be the motive associated with the modular form, and let $H^1_{\operatorname{mot}}(\mathbb{Q},\mathcal{M})$ be its motivic cohomology. For a fixed $\mathbb{Q}$ -basis $\mathscr{B}$ , let $\operatorname{Reg}_{\mathscr{B}}(\mathcal{M})$ denote the determinant of the Gillet–Soulé height pairing.

Gillet–Soulé non-degeneracy conjecture. The Gillet–Soulé height pairing is non-degenerate.

This is one of the three conjectural inputs assumed before the paper's final fractional-ideal formula. The excerpt gives no resolution status.
source: Enrico Da Ronche, “Kolyvagin's conjecture for modular forms at non-ordinary primes”, arXiv:2503.09955 (2025).

References

Primary source

Matteo Longo and Stefano Vigni, “The Tamagawa number conjecture and Kolyvagin's conjecture for motives of modular forms”, arXiv:2211.04907 (2023).

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