Extend the Moret–Bailly key formula to general adelic curves

Establish the Moret–Bailly key formula for the adelic vector bundle of global sections of a symmetric line bundle on an abelian scheme over a general adelic curve, including the identity relating its average slope to the Faltings height.

Background

The paper uses the Moret–Bailly key formula, in Bost’s reformulation, to relate the average slope of the adelic vector bundle of global sections of a symmetric ample line bundle to the Faltings height of an abelian variety. This relation is available in the number-field and function-field settings and is central to the standard proof of the Bogomolov conjecture for divisors.

For a general adelic curve, the authors do not establish the analogous identity. They indicate that proving it would require understanding compactifications of symmetric line bundles on abelian schemes over quasi-projective bases. The paper instead handles the almost-everywhere good-reduction case through Tang’s determinant isomorphism, so the general formula remains unresolved.

References

For a general countable GVF $K$, viewing $K$ as an adelic curve instead, we may view $\pi_* #1{L}$ as an adelic vector bundle in the sense of Chen--Moriwaki Theorem 6.1.13. However, the formula (\ref{eq: MoretBailly1}) is not known to hold in the setting of a general adelic curve and indeed, it seems that would require understanding various compactifications of symmetric line bundles $\mathcal{L}$ on an abelian scheme $\mathcal{A}/S$ when $S$ is only quasi-projective (see for the case of the principal theta divisor).

— Uniform Height Gaps in Arbitrary Characteristics  (2610.11905 - Lin et al., 8 Oct 2026) in Section 6, “Bogomolov Conjecture for Divisors over GVFs”