Positive definiteness of Gillet–Soulé height pairings

Prove that the Gillet–Soulé archimedean height pairing on the relevant Heegner-type cycles in the Kuga–Sato varieties is positive definite for every even weight specialization satisfying the congruence condition κ ≡ k (mod p − 1).

Background

Positive definiteness of the height pairing is used to convert the nontriviality of specialized Heegner cycles into the nonvanishing of central derivatives of Rankin–Selberg L-functions. The required assertion is imposed as Assumption 3.20 for all relevant even-weight specializations.

The paper explains that this assertion is a special case of conjectural arithmetic analogues of the standard conjectures, as well as of broader conjectures of Beilinson and Bloch on positive definiteness of height pairings. It also states that no result establishing the condition is known in the higher-weight setting considered.

References

The positive definiteness of ⟨·, ·⟩κ,GS is a consequence of one of the arithmetic analogues of the standard conjectures proposed by Gillet and Soulé ([8, Conjecture 2]); it is also a special case of general conjectures of Beilinson ([1]) and Bloch ([2]) on positive definiteness of height pairings. While it is natural to impose such a positive definiteness condition when studying the arithmetic of Heegner cycles (see, e.g., [40, Assumption 4.1]), we are not aware of any result in this direction in our higher weight setting.

— Analytic rank one propagation in Hida families  (2608.13145 - Vigni, 13 Aug 2026) in Assumption 3.20 and Remark 3.21, Section 3.6, p. 12