Symmetric roots and admissible pairing

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.