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We develop a theory of $p$-adic N\'eron functions on abelian varieties,depending on various auxiliary choices, and show that the global $p$-adicheight functions constructed by Mazur and Tate can be decomposed into a sum of$p$-adic N\'eron functions if the same auxiliary choices are made. We alsodecompose the $p$-adic height constructed by Coleman and Gross, and extended toarbitrary reduction by Colmez and Besser, into a sum of local height functionsfor Jacobians of odd degree genus $2$ curves. We show that this local heightfunction is equal to the $p$-adic N\'eron function with the same auxiliarychoices, regardless of the reduction type of the curve. This extends work ofBalakrishnan and Besser for elliptic curves. When the curve has semistablereduction and the reduction of the Jacobian is ordinary, we also describe the$p$-adic N\'eron function that arises from the canonical Mazur-Tate splittingexplicitly in terms of a generalisation of the $p$-adic sigma functionconstructed by Blakestad.

Coleman-Gross Heights and $p$-adic N\'eron Functions on Jacobians of Genus $2$ Curves