Néron-Tate heights of cycles on jacobians

Abstract

We develop a method to calculate the Néron-Tate height of tautological integral cycles on jacobians of curves defined over number fields. As examples we obtain closed expressions for the Néron-Tate height of the difference surface, the Abel-Jacobi images of the square of the curve, and of any symmetric theta divisor. As applications we obtain a new effective positive lower bound for the essential minimum of any Abel-Jacobi image of the curve and a proof, in the case of jacobians, of a formula proposed by Autissier relating the Faltings height of a principally polarized abelian variety with the Néron-Tate height of a symmetric theta divisor.

Faltings height and Néron–Tate height of a theta divisor

“…Using (1.10) and Theorem B, the formula from Theorem A specializes into the formula for the stable Faltings height of . This recovers [dJ18, Theorem 1.6].…”

“…In [Aut06], Autissier established the identity (1.4) for elliptic curves, for Jacobians of genus two curves and for arbitrary products of these. In [dJ18, Theorem 1.6], the first-named author exhibited natural , and established (1.4), for all Jacobians and for arbitrary products of these. In both [Aut06, dJ18], the local non-archimedean invariants are expressed in terms of the combinatorics of the dual graph of the underlying semistable curve at .…”

Arithmetic bigness and a uniform Bogomolov-type result

“…For a more precise comparison, recall that the original Bogomolov conjecture was proved by Ullmo [Ull] in terms of the equidistribution theorem of [SUZ], and a second proof in terms of bounding the self-intersection number of the admissible canonical bundle was obtained along the line of Zhang [Zha1,Zha3], Cinkir [Cin] and de Jong [dJo2]. Then the treatment of [Kuh] is a family version of that of [SUZ, Ull], and our treatment is a family version of that of [Zha1,Zha3,Cin,dJo2].…”

“…Now we sketch our proof of Theorem 1.3. The process is to align the relevant results of [Zha1,Zha3,Cin,dJo2] into a family. With some effort, we can reduce it to the statement that π * ω X/S,a , ω X/S,a is nef and big over S.…”

On the height of Gross–Schoen cycles in genus three

“…We mention that in [ 27 ], Corollary 1.4] it is shown unconditionally that the inequality holds. This inequality is weaker than ( 4.1 ) if but still implies the Bogomolov conjecture for X .…”