Robin de Jong's home page

Preprints | Publications in journals | Contributions to books | Books edited | Other writings

My research deals with arithmetic and analytic aspects of curves and abelian varieties and their moduli spaces. Most of my research is inspired by, and formulated in terms of, Arakelov theory, which is a geometrical approach to Diophantine problems. More specifically I am interested in the behavior of hermitian metrics of Arakelovian nature over moduli spaces, especially under degeneration towards the boundary.

In the case of curves, boundary objects are described and studied using tools from graph theory and electric network theory; in the case of abelian varieties, boundary objects are described and studied using tools from the theory of lattices. Many intricate and fundamental connections have arisen in this way between graphs, lattices, and moduli spaces, and we continue to find more.

Contact
Cv

Publications on:

arXiv
ORCiD
Google Scholar
MathSciNet
zbMATH

Preprints:

Pure extension of the theta divisor over the moduli space of abelian varieties (joint with Ana Botero, José Burgos Gil and David Holmes). arxiv
Integral torsion points on abelian varieties over function fields (joint with Nicole Looper and Farbod Shokrieh). arxiv
Canonical local heights and Berkovich skeleta (joint with Farbod Shokrieh). arxiv

Publications in journals:

Rings of Siegel-Jacobi forms of bounded relative index are not finitely generated (joint with Ana Botero, José Burgos Gil and David Holmes).
Duke Math. Jnl. 173 (2024), 2315--2396. arxiv | journal
Jumps in the height of the Ceresa cycle (joint with Farbod Shokrieh).
Jnl. Differential Geom. 126 (2024), 169--214. arxiv | journal
Rings of tautological forms on moduli spaces of curves (joint with Stefan van der Lugt).
Expo. Math. 41 (2023), 531--565. arxiv | journal
Computing heights via limits of Hodge structures (joint with Spencer Bloch and Emre Sertöz).
Experimental Math. 33 (2024) arxiv | journal
Heights on curves and limits of Hodge structures (joint with Spencer Bloch and Emre Sertöz).
J. London Math. Soc. 108 (2023), 340--361. arxiv | journal
Chern-Weil and Hilbert-Samuel formulae for singular hermitian line bundles (joint with Ana Botero, José Burgos Gil and David Holmes).
Doc. Math. 27 (2022), 2563-2623. arxiv | journal
Tropical moments of tropical Jacobians (joint with Farbod Shokrieh).
Canadian Jnl. Math. (2022), 1--31. arxiv | journal
Metric graphs, cross ratios, and Rayleigh's laws (joint with Farbod Shokrieh).
Rocky Mountain Jnl. Math. 52 (2022), no. 4, 1403--1422. arxiv | journal
Faltings height and Néron-Tate height of a theta divisor (joint with Farbod Shokrieh).
Compositio Math. 158 (2022), no.1, 1--32. arxiv | journal
Frobenius' theta function and Arakelov invariants in genus three.
Pure and Applied Math. Quarterly 16 (2020), no.5, 1387--1418. arxiv | journal
Chern-Weil theory for line bundles with the family Arakelov metric (joint with Michiel Jespers).
Michigan Math. J. 69 (2020), no. 1, 3--40. arxiv | journal
Faltings delta-invariant and semistable degeneration.
Jnl. Differential Geom. 111 (2019), 241--301. arxiv | journal
On the height of Gross-Schoen cycles in genus three.
Research in Number Theory 4:38 (2018). arxiv | journal
Singularities of the biextension metric for families of abelian varieties (joint with José Burgos Gil and David Holmes).
Forum Math. Sigma 6, e12 (2018). arxiv | journal
Néron-Tate heights of cycles on jacobians.
J. Alg. Geom. 27 (2018), 339--381. arxiv | journal
Positivity of the height jump divisor (joint with José Burgos Gil and David Holmes).
IMRN, rnx169, doi.org/10.1093/imrn/rnx169 (2017). arxiv | journal
Néron models and the height jump divisor (joint with Owen Biesel and David Holmes).
Trans. AMS 369 (2017), 8685--8723. arxiv | journal
Point-like limit of the hyperelliptic Zhang-Kawazumi invariant.
Pure Appl. Math. Q. 11 (2015), 633--653. arxiv | journal
Asymptotics of the Néron height pairing (joint with David Holmes).
Math. Res. Lett. 22 (2015), 1337--1371. arxiv | journal
Canonical heights and division polynomials (joint with J. Steffen Müller).
Math. Proc. Camb. Phil. Soc. 157 (2014), 357--373. arxiv | journal
Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces.
Asian Jnl. Math. 18 (2014), 507--524. arxiv | journal
Normal functions and the height of Gross-Schoen cycles.
Nagoya Math. Jnl. 213 (2014), 53--77. arxiv | journal
Second variation of Zhang's λ-invariant on the moduli space of curves.
American Jnl. Math. 135 (2013), 275--290. arxiv | journal
Special values of canonical Green's functions.
Math. Z. 274 (2013), 199--208. arxiv | journal
Conjecture de Shafarevitch effective pour les revêtements cycliques (joint with Gaël Rémond).
Algebra and Number Theory 5 (2011), 1133--1143. journal
Local heights on Galois covers of the projective line.
Acta Arithm. 152 (2012), 51--70. arxiv | journal
One half log discriminant and division polynomials.
Archiv Math. 97 (2011), 251--257. arxiv | journal
Symmetric roots and admissible pairing.
Trans. AMS 363 (2011), 4263--4283. arxiv | journal
Admissible constants for genus 2 curves.
Bull. London Math. Soc. 42 (2010), 405--411. arxiv | journal
Covers of surfaces with fixed branch locus (joint with Bas Edixhoven and Jan Schepers).
Int. Jnl. Math. 21 (2010), 859--874. arxiv | journal
Local invariants attached to Weierstrass points.
Manuscripta Math. 129 (2009), 273--292. arxiv | journal
Theta functions on the theta divisor.
Rocky Mountain Jnl. Math. 40 (2010), 155--176. arxiv | journal
Explicit Mumford isomorphism for hyperelliptic curves.
Jnl. Pure Appl. Algebra 208 (2007), 1--14. arxiv | journal
Arakelov invariants of Riemann surfaces.
Doc. Math. 10 (2005), 311--329. arxiv | journal
On the Arakelov theory of elliptic curves.
l'Ens. Math. 51 (2005), 179--201. arxiv | journal

Contributions to books:

Torus bundles and 2-forms on the universal family of Riemann surfaces.
In: A. Papadopoulos (ed.), Handbook of Teichmüller theory. Volume VI. EMS Publishing House 2016. arxiv | book
Short introduction to heights and Arakelov theory (joint with Bas Edixhoven).
In: J.-M. Couveignes, B. Edixhoven (eds.), Computational aspects of Modular Forms and Galois Representations. Annals of Mathematics Studies 176, Princeton University Press 2011. arxiv | book
Applying Arakelov theory (joint with Bas Edixhoven).
In: J.-M. Couveignes, B. Edixhoven (eds.), Computational aspects of Modular Forms and Galois Representations. Annals of Mathematics Studies 176, Princeton University Press 2011. arxiv | book
Bounds for Arakelov invariants of modular curves (joint with Bas Edixhoven)
In: J.-M. Couveignes, B. Edixhoven (eds.), Computational aspects of Modular Forms and Galois Representations. Annals of Mathematics Studies 176, Princeton University Press 2011. arxiv | book
Gauss map on the theta divisor and Green's functions.
In: B. Edixhoven, G. van der Geer and B. Moonen (eds.), Modular Forms on Schiermonnikoog. Cambridge University Press 2008. arxiv | book
Faltings' delta-invariant of a hyperelliptic Riemann surface.
In: G. van der Geer, B. Moonen, R. Schoof (eds.), Number Fields and Function Fields -- Two Parallel Worlds. Progress in Mathematics vol. 239, Birkhäuser Verlag 2005. arxiv | book

Books edited:

Geometry and Arithmetic.
C. Faber, G. Farkas, R. de Jong (eds.), EMS Series of Congress Reports, EMS Publishing House, 2012. book

Other writings:

In memoriam Bas Edixhoven: Genereus, optimistisch en betrokken.
(joint with G. van der Geer, R. Schoof and J. Top). Nieuw Archief voor Wiskunde 23 (2022), 201--207. journal
Explicit Arakelov geometry.
PhD thesis, University of Amsterdam, December 2004. book