arXiv subjects

Evan M. O'Dorney

Publications and source records attributed to Evan M. O'Dorney.

20 records · Page 2Linked to original sources

arXiv2016

A remarkable identity in class numbers of cubic rings

In 1997, Y. Ohno empirically stumbled on an astoundingly simple identity relating the number of cubic rings $h(D)$ of a given discriminant $D$, over the integers, to the number of cubic rings $h'(D)$ of discriminant $-27D$ in which every element has trace divisible by 3: $h'(D) = h(D)$, if $D < 0$, and $h'(D) = 3h(D)$ if $D > 0$, where in each case, rings are weighted by the reciprocal of their number of automorphisms. This allows the functional equations governing the analytic continuation of the Shintani zeta functions (the Dirichlet series built from the values of $h$ and $h'$) to be put in self-reflective form. In 1998, J. Nakagawa verified Ohno's conjecture. We present a new proof of the identity that uses the main ingredients of Nakagawa's proof (binary cubic forms, recursions, and class field theory), as well as one of Bhargava's celebrated higher composition laws, while aiming to stay true to the stark elegance of the identity.

math.NT↗

arXiv2014

Canonical rings of Q-divisors on P^1

The canonical ring $S_D = \bigoplus_{d \geq 0} H^0(X, \lfloor dD \rfloor)$ of a divisor D on a curve X is a natural object of study; when D is a Q-divisor, it has connections to projective embeddings of stacky curves and rings of modular forms. We study the generators and relations of S_D for the simplest curve X = P^1. When D contains at most two points, we give a complete description of S_D; for general D, we give bounds on the generators and relations. We also show that the generators (for at most five points) and a Groebner basis of relations between them (for at most four points) depend only on the coefficients in the divisor D, not its points or the characteristic of the ground field; we conjecture that the minimal system of relations varies in a similar way. Although stated in terms of algebraic geometry, our results are proved by translating to the combinatorics of lattice points in simplices and cones.

math.AG↗