In this paper we prove new explicit formulas for Faltings' $δ$-invariant of an arbitrary hyperelliptic Riemann surface. This has several applications: For example we obtain an explicit lower bound for $δ$ depending only on the genus, and we deduce new explicit bounds for the Arakelov self-intersection number $ω^2$ associated to hyperelliptic curves over number fields. Furthermore, we obtain an improved version of Szpiro's small points conjecture for hyperelliptic curves of genus at least $3$. Our method allows us in addition to establish a generalization of Rosenhain's formula on $θ$-derivativ…

2016-05-04math.AGmath.NT

New explicit formulas for Faltings' delta-invariant

In this paper we give new explicit formulas for Faltings' $δ$-invariant in terms of integrals of theta functions, and we deduce an explicit lower bound for $δ$ only in terms of the genus and an explicit upper bound for the Arakelov-Green function in terms of $δ$. Furthermore, we give a canonical extension of $δ$ and the Zhang-Kawazumi invariant $φ$ to the moduli space of indecomposable principally polarised complex abelian varieties.

2016-12-03math.AGmath.NT

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.

2017-10-17math.AG

A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces

The Kawazumi-Zhang invariant $φ$ for compact genus-two Riemann surfaces was recently shown to be a eigenmode of the Laplacian on the Siegel upper half-plane, away from the separating degeneration divisor. Using this fact and the known behavior of $φ$ in the non-separating degeneration limit, it is shown that $φ$ is equal to the Theta lift of the unique (up to normalization) weak Jacobi form of weight $-2$. This identification provides the complete Fourier-Jacobi expansion of $φ$ near the non-separating node, gives full control on the asymptotics of $φ$ in the various degeneration limits, and p…

2016-02-09hep-thmath.AG

Polynomial bounds for Arakelov invariants of Belyi curves

We explicitly bound the Faltings height of a curve over Q polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings' delta invariant and the self-intersection of the dualizing sheaf. Our results allow us to explicitly bound Arakelov invariants of modular curves, Hurwitz curves and Fermat curves in terms of their genus. Moreover, as an application, we show that the Couveignes-Edixhoven-Bruin algorithm to compute coefficients of modular forms for congruence subgroups of SL2(Z) runs in polynomial time under the Riemann hypothesis f…

2014-03-25math.AGmath.NT

Local invariants attached to Weierstrass points

Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation of the hyperelliptic discriminant of X/S, and the valuation of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable these invariants are known to satisfy certain inequalities. We prove an exact formula relating the two invariants with intersection theoretic data determined by the distribution of Weierstrass points over the special fiber, in the sem…

2012-04-24math.AG

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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.

2004-08-25math.AGmath.NT

Frobenius' theta function and Arakelov invariants in genus three

We give explicit formulas for the Kawazumi-Zhang invariant and Faltings delta-invariant of a compact and connected Riemann surface of genus three. The formulas are in terms of two integrals over the associated jacobian, one integral involving the standard Riemann theta function, and another involving a theta function particular to genus three that was discovered by Frobenius. We review part of Frobenius' work on his theta function and connect our results with a formula due to Bloch, Hain and Bost describing the archimedean height pairing of Ceresa cycles in genus three.

2020-01-31math.AGmath.NT

Torus bundles and 2-forms on the universal family of Riemann surfaces

We revisit three results due to Morita expressing certain natural integral cohomology classes on the universal family of Riemann surfaces C_g, coming from the parallel symplectic form on the universal jacobian, in terms of the Miller-Morita-Mumford classes e and e_1. Our discussion will be on the level of the natural 2-forms representing the relevant cohomology classes, and involves a comparison with other natural 2-forms representing e, e_1 induced by the Arakelov metric on the relative tangent bundle of C_g over M_g. A secondary object called a_g occurs, which was discovered and studied by K…

2014-01-06math.AGmath.GT