I have been interested in K3 surfaces for a long time. Formerly I published papers on the Brauer-Manin obstruction, but in recent years my interest has shifted in the direction of modularity and relations between K3 surfaces and curves. I retain an interest in the use of moduli theory and Mori theory in arithmetic, although I have not published on moduli of curves in many years. Here are links to my papers.
A variant of the Erdős-Rényi random graph process} (with M. Molloy and P. Pra\l at), J. Graph Theory 102 (2) (2023), 322--345, arXiv.
Definite orthogonal modular forms: computations, excursions, and discoveries (with E. Assaf, D. Fretwell, C. Ingalls, S. Secord, and J. Voight), to appear in the Proceedings of ANTS XV, arXiv.
Explicit coverings of families of elliptic surfaces by squares of curves (with C. Ingalls and O. Patashnick), Math. Z. 302 (2) (2022), 1191--1238, arXiv.
Modularity of two double covers of P5 branched along 12 hyperplanes, Ramanujan J. (2022), link.
Quotient graphs and amalgam presentations for unitary groups over cyclotomic rings (with C. Ingalls, B. Jordan, A. Keeton, and Y. Zaytman), Exp. Math. (2021), link.
On the Cynk-Hulek criterion for crepant resolutions of double covers}(with C. Ingalls), J. Pure Appl. Algebra (2022), link.
The Clifford-cyclotomic group and Euler-Poincar\'e characteristics (with C. Ingalls, B. Jordan, A. Keeton, and Y. Zaytman), Can. Math. Bull. 64} (3) (2021), 651--666, link.
Appendix to the paper The rings of Hilbert modular forms for Q(sqrt(29)) and Q(sqrt(37)), by B. Williams, J. Algebra 559 (2020), 679--711, link.
Finiteness theorems for K3 surfaces over arbitrary fields (with M. Bright and R. van Luijk), Eur. J. Math. 6 (2020), 336--366, link.
New realizations of modular forms in Calabi-Yau threefolds arising from phi4 theory, J. Number Theory184 (2018), 342--383, link.
The two faces of the twisted Kummer surface, Acta Arith.143 (2010), 1--20, link.
Density of rational points on diagonal quartic surfaces (with D. McKinnon and R. van Luijk), Algebra Number Theory 4 (2010), 1--20, link.
Nontrivial elements of Sha explained through K3 surfaces (with R. van Luijk), Math. Comput. 78 (2009), 441--483, link.
The Brauer-Manin obstruction on del Pezzo surfaces of degree 2 branched along a plane section of a Kummer surface, Math. Proc. Camb. Phil. Soc. 144 (2008), 623--638, link.
Asymptotics of coinvariants of Iwasawa modules under non-normal subgroups (with J. Ellenberg), Math. Res. Lett. 14 (2007), 769--773, link.
The Brauer-Manin Obstruction and Sha[2] (with M. Bright, N. Bruin, and E. V. Flynn), LMS J. Comput. Math. 10 (2007), 354--377, link.
Heegner points and Hilbert modular forms} (with Henri Darmon), appearing in: Algebraic number theory and related topics (Japanese) (Kyoto, 2002). Surikaisekikenkyusho Kokyuroku No. 1324 (2003),153--160. (Summary of results in the next item.)
Periods of Hilbert modular forms and rational points on elliptic curves (with Henri Darmon). Int. Math. Res. Not. 2003, no. 40, 2153--2180, link.
Relations among divisors on the moduli space of curves with marked points, Comm. Algebra 31 (2003), no. 3, 1175--1185, link.
The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125 (2003), no. 1, 105--138, link.
Deformations of Galois representations arising from degenerate extensions, Acta Arith. 101 (2002), no. 3, 293--302, link.
Kodaira dimensions of moduli spaces of curves with marked points, doctoral thesis.
The basic theory of Clifford-Bianchi groups for hyperbolic n-space (with T. Dupuy, A. Hildo, and C. Ingalls), arXiv, submitted.
The corank of unitary groups over cyclotomic rings (with C. Ingalls, B. Jordan, A. Keeton, and Y. Zaytman), arXiv, to appear in J. London Math. Soc.
The Zariski closure of integral points on varieties parametrizing periodic continued fractions (with B. Jordan and Y. Zaytman), arXiv, submitted.
Higher modularity of elliptic curves over function fields (with J. Weinstein, with an appendix by M. Kuwata), arXiv, submitted.
Rank-based linkage I: triplet comparisons and oriented simplicial complexes (with R. Darling and W. Grilliette), arXiv, submitted.
The automorphisms of certain Kummer surfaces act on the points mod p by the alternating group} (with O. Patashnick), arXiv, submitted.
Rings of Hilbert modular forms, computations on Hilbert modular surfaces, and the Oda-Hamahata conjecture, arXiv, submitted.
Certifying nontriviality of Ceresa classes of curves (with J. Ellenberg and P. Srinivasan), arXiv.
Orthogonal modular forms attached to quaternary lattices} (with E. Assaf, D. Fretwell, C. Ingalls, S. Secord, and J. Voight).
Surfaces and threefolds parametrizing periodic continued fractions (with B. Jordan and Y. Zaytman).
Modular forms and rigid surfaces of geometric genus 2 (with A. Auel and J. Voight).
Results on SL2 of orders in quaternion algebras (with T. Dupuy).
Hecke operators and modular symbols for Clifford modular forms (with T. Dupuy).
Constructing rigid Calabi-Yau threefolds from families of K3 surfaces.
Modular forms and rigid surfaces of geometric genus 2 (with A. Auel and J. Voight).
A category of mixed abelian motives for very general abelian varieties (with O. Patashnick and A. Torszewski).