On the height of Gross-Schoen cycles in genus three

We show that there exists a sequence of genus three curves defined over the rationals in which the height of a canonical Gross-Schoen cycle tends to infinity.Comment: 26 pages; v2: referee's remarks taken into accoun

N\'eron-Tate heights of cycles on jacobians

We develop a method to calculate the N\'eron-Tate height of tautological integral cycles on jacobians of curves defined over number fields. As examples we obtain closed expressions for the N\'eron-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 N\'eron-Tate height of a symmetric theta divisor.Comment: 35 pages, SAGE file written by David Holmes is available as an ancillary file, v2: minor revision

Local heights on Galois covers of the projective line

Let X be a smooth projective curve of positive genus defined over a number field K. Assume given a Galois covering map x from X to the projective line over K and a place v of K. We introduce a local canonical height on the set of K_v-valued points of X associated to x as an integral with logarithmic integrand, generalizing Tate's local Neron function on an elliptic curve. The resulting global height can be viewed as a 'Mahler measure' associated to x. We prove that the local canonical height can be obtained by averaging, and taking a limit, over divisors of higher order Weierstrass points on X. This generalizes previous results by Everest-ni Fhlathuin and Szpiro-Tucker. Our construction of the local canonical height is an application of potential theory on Berkovich curves in the presence of a canonical measure.Comment: 18 page

Arakelov invariants of Riemann surfaces

We derive explicit formulas for the Arakelov-Green function and the Faltings delta-invariant of a Riemann surface. A numerical example illustrates how these formulas can be used to calculate Arakelov invariants of curves.Comment: 16 pages; 2nd version. Same main results as in 1st version, but with a numerical example added. Section 8 of 1st version is now incorporated in new article ``Faltings' delta-invariant of a hyperelliptic Riemann surface'

Faltings delta-invariant and semistable degeneration

We determine the asymptotic behavior of the Arakelov metric, the Arakelov-Green's function, and the Faltings delta-invariant for arbitrary one-parameter families of complex curves with semistable degeneration. The leading terms in the asymptotics are given a combinatorial interpretation in terms of S. Zhang's theory of admissible Green's functions on polarized metrized graphs.Comment: 50 page