Discrete and metric divisorial gonality can be different - to appear in J. Comb. Theory. Ser. A
(joint with Josse van Dobben de Bruyn and Marieke van der Wegen)
Abstract: This paper compares the divisorial gonality of a finite graph G to the divisorial gonality of the associated graph. We show that the divisorial gonality of the metric graph is equal to the minimal divisorial gonality of all regular subdivisions of G, and we provide a class of graphs for which this number is strictly smaller than the divisorial gonality of G. This settles a conjecture of M. Baker in the negative.
Links: ArXiv
The 4-rank of class groups of K(\sqrt(n)) - preprint
(joint with Peter Koymans and Adam Morgan)
Abstract: Let K/Q be a quadratic extension. In this paper we study the 4-rank of the class group Cl(K(\sqrt(n)), where n varies over the squarefree rational integers. We show that for 100% of squarefree n, the 4-rank is given by an explicit formula involving the 2-rank of Cl(K) and the number of prime factors of n which are inert in K/Q.
Links: ArXiv
Constructing Tree Decompositions of Graphs with Bounded Gonality - COCOON 2020
(joint with Hans L. Bodlaender, Josse van Dobben de Bruyn and Dion Gijswijt)
Abstract: We give a constructive proof of the fact that the treewidth of a graph is at most its divisorial gonality. The proof gives a polynomial time algorithm to construct a tree decomposition of width at most k, when an effective divisor of degree k that reaches all vertices is given. We also give a similar result for two related notions: stable divisorial gonality and stable gonality.
Links: ArXiv
L-series and homomorphisms of number fields - preprint
Abstract: While the zeta function does not determine a number field uniquely, the L-series of a well-chosen Dirichlet character does. Moreover, isomorphisms between two number fields are in natural bijection with L-series preserving isomorphisms of l-torsion subgroups of the Dirichlet character groups. We extend this by showing that homomorphisms between number fields are in natural bijection with group homomorphisms between l-torsion subgroups of the Dirichlet character groups abiding a divisibility condition on the L-series when l is sufficiently large.
Links: ArXiv
L-series and isogenies of abelian varieties - preprint
Abstract: Faltings's isogeny theorem states that two abelian varieties are isogenous over a number field precisely when the characteristic polynomials of the reductions at almost all prime ideals of the number field agree. This implies that two abelian varieties over Q with the same L-series are necessarily isogenous, but this is false over a general number field. Let A and A′ be two abelian varieties, defined over number fields K and K′ respectively. Our main result is that A and A′ are isogenous after a suitable isomorphism between K and K′ if and only if the Dirichlet character groups of K and K′ are isomorphic and the L-series of A and A′ twisted by the Dirichlet characters match.
Links: ArXiv
L-series and isomorphisms of number fields - to appear in J. Numb. Th.
Abstract: Two number fields with equal Dedekind zeta function are not necessarily isomorphic. However, if the number fields have equal sets of Dirichlet L-series then they are isomorphic. We extend this result by showing that the isomorphisms between the number fields are in bijection with L-series preserving isomorphisms between the character groups.
Links: ScienceDirect | ArXiv
Reconstructing global fields from dynamics in the abelianized Galois group - Selecta Math. (N. S.), 25, 24, 2019
(joint with Gunther Cornelissen, Xin Li and Matilde Marcolli)
Abstract: We study a dynamical system induced by the Artin reciprocity map for a global field. We translate the conjugacy of such dynamical systems into various arithmetical properties that are equivalent to field isomorphism, relating it to anabelian geometry.
Links: SpringerLink | ArXiv
Characterization of global fields by Dirichlet L-series - Res. in Numb. Th., 5, 7, 2019
(joint with Gunther Cornelissen, Bart de Smit, Xin Li and Matilde Marcolli)
Abstract: We prove that two global fields are isomorphic if and only if there is an isomorphism of groups of Dirichlet characters that preserves L-series.
Links: SpringerLink | ArXiv
Computing graph gonality is hard - Discrete Appl. Math., 287, pp 134-149, 2020
(joint with Dion Gijswijt and Marieke van der Wegen)
Abstract: There are several notions of gonality for graphs. The divisorial gonality dgon(G) of a graph G is the smallest degree of a divisor of positive rank in the sense of Baker-Norine. The stable gonality sgon(G) of a graph G is the minimum degree of a finite harmonic morphism from a refinement of G to a tree, as defined by Cornelissen, Kato and Kool. We show that computing dgon(G) and sgon(G) are NP-hard by a reduction from the maximum independent set problem and the vertex cover problem, respectively. Both constructions show that computing gonality is moreover APX-hard.
Links: ScienceDirect | Arxiv