Divisors on graphs, Connected flags, and Syzygies

We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gr\"obner theory. We give an explicit description of a minimal Gr\"obner bases for each higher syzygy module. In each case the given minimal Gr\"obner bases is also a minimal generating set. The Betti numbers of the binomial ideal and its natural initial ideal coincide and they correspond to the number of 'connected flags' in the graph. In particular the Betti numbers are independent of the characteristic of the base field. For complete graphs the problem was previously studied by Postnikov and Shapiro and by Manjunath and Sturmfels. The case of a general graph was stated as an open problem.Comment: to appear in International Mathematics Research Notices (IMRN

Faltings height and N\'eron-Tate height of a theta divisor

We prove a formula which, given a principally polarized abelian variety $(A,\lambda)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the N\'eron-Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. We introduce the notion of a tautological model of a principally polarized abelian variety over a complete discretely valued field, and we express the non-archimedean N\'eron functions for a symmetric theta divisor on $A$ in terms of tautological models and the tropical Riemann theta function.Comment: Referee's remarks taken into account, errors fixed, main result unchanged, added examples dealing with elliptic curve

The monodromy pairing and discrete logarithm on the Jacobian of finite graphs

Every graph has a canonical finite abelian group attached to it. This group has appeared in the literature under a variety of names including the sandpile group, critical group, Jacobian group, and Picard group. The construction of this group closely mirrors the construction of the Jacobian variety of an algebraic curve. Motivated by this analogy, it was recently suggested by Norman Biggs that the critical group of a finite graph is a good candidate for doing discrete logarithm based cryptography. In this paper, we study a bilinear pairing on this group and show how to compute it. Then we use this pairing to find the discrete logarithm efficiently, thus showing that the associated cryptographic schemes are not secure. Our approach resembles the MOV attack on elliptic curves