  Using the discriminant modular form and the Noether formula it is possible to
write the admissible self-intersection of the relative dualising sheaf of a
semistable hyperelliptic curve over a number field or function field as a sum,
over all places, of a certain adelic invariant. We provide a simple geometric
interpretation for this invariant, based on the arithmetic of symmetric roots.
We propose the conjecture that the invariant introduced in this paper coincides
with an invariant introduced in a recent paper by S.-W. Zhang.
