Symmetric roots and admissible pairing

de Jong, Robin

doi:10.1090/S0002-9947-2011-05217-7

Symmetric roots and admissible pairing
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Trans. Amer. Math. Soc. 363 (2011), 4263-4283 Request permission

Abstract:

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 $\chi$. We provide a simple geometric interpretation for this invariant $\chi$, based on the arithmetic of symmetric roots. We propose the conjecture that the invariant $\chi$ coincides with the invariant $\varphi$ introduced in a recent paper by S.-W. Zhang. This conjecture is true in the genus $2$ case, and we obtain a new proof of the Bogomolov conjecture for curves of genus $2$ over number fields.

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