On arithmetic intersection numbers on self-products of curves

Abstract

We give a close formula for the Néron-Tate height of tautological integral cycles on Jacobians of curves over number fields as well as a new lower bound for the arithmetic self-intersection number ω2 of the dualizing sheaf of a curve in terms of Zhang's invariant ϕ. As an application, we obtain an effective Bogomolov-type result for the tautological cycles. We deduce these results from a more general combinatorial computation of arithmetic intersection numbers of adelic line bundles on higher self-products of curves, which are linear combinations of pullbacks of line bundles on the curve and the diagonal bundle.

Degeneration of Riemann theta functions and of the Zhang-Kawazumi invariant with applications to a uniform Bogomolov conjecture

“…In [24] we computed h ′ L (Z m,α ) in terms of ω2 X , ϕ(X) and h ′ L α − 1 2g−2 ω X . Combining this result with the bound in Corollary 1.4, we are able to deduce the following uniform bound for the Bogomolov conjecture in this case.…”

“…j (x j − α)and we write Z m,α for the cycle in J obtained by the image of f m,α . In[24, Theorem 1.1] we computed h ′ L (Z m,α ) and we may deduce from this resulth ′ L (Z m,α ) ≥ g−r 2dK g−1) 2 − (2g+1) r j<k mj m k 6g(g−1) 2 (g−2) ω2 X + r j<k mj m k 3g(g−1)(g−2) ϕ(X) ,(8.4)where for r = 1 the fractions with factor r j<k m j m k = 0 in the nominator are also set to be 0 if g = 2. This inequality allows us to prove Corollary 1.5.Proof of Corollary 1.5.…”