9 research outputs found
Quintic algebras over Dedekind domains and their resolvents
In this paper, the parametrization of rings of ranks 2, 3, and 4 over
Dedekind domains, found in the author's paper "Rings of small rank over a
Dedekind domain and their ideals," is extended to rank 5, following Bhargava's
parametrization of quintic Z-algebras.Comment: 10 page
Rings of small rank over a Dedekind domain and their ideals
In 2001, M. Bhargava stunned the mathematical world by extending Gauss's
200-year-old group law on integral binary quadratic forms, now familiar as the
ideal class group of a quadratic ring, to yield group laws on a vast assortment
of analogous objects. His method yields parametrizations of rings of degree up
to 5 over the integers, as well as aspects of their ideal structure, and can be
employed to yield statistical information about such rings and the associated
number fields.
In this paper, we extend a selection of Bhargava's most striking
parametrizations to cases where the base ring is not Z but an arbitrary
Dedekind domain R. We find that, once the ideal classes of R are properly
included, we readily get bijections parametrizing quadratic, cubic, and quartic
rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss
composition for which Bhargava is famous. We expect that our results will shed
light on the analytic distribution of extensions of degree up to 4 of a fixed
number field and their ideal structure.Comment: 39 pages, 1 figure. Harvard College senior thesis, edite
Continued Fractions and Linear Fractional Transformations
Rational approximations to a square root can be produced by
iterating the transformation starting from for
any positive integer . We show that these approximations coincide infinitely
often with continued fraction convergents if and only if is an
integer, in which case the continued fraction has a rich structure. It consists
of the concatenation of the continued fractions of certain explicitly definable
rational numbers, and it belongs to one of infinitely many families of
continued fractions whose terms vary linearly in two parameters. We also give
conditions under which the orbit consists exclusively of
convergents or semiconvergents and prove that with few exceptions it includes
all solutions to the Pell equation .Comment: 18 page
Visibly irreducible polynomials over finite fields
H. Lenstra has pointed out that a cubic polynomial of the form
(x-a)(x-b)(x-c) + r(x-d)(x-e), where {a,b,c,d,e} is some permutation of
{0,1,2,3,4}, is irreducible modulo 5 because every possible linear factor
divides one summand but not the other. We classify polynomials over finite
fields that admit an irreducibility proof with this structure.Comment: 11 pages. To appear in the American Mathematical Monthl
Canonical rings of Q-divisors on P^1
The canonical ring 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.Comment: 19 pages, 3 figure
Large orbits on Markoff-type K3 surfaces over finite fields
We study the surface in , a tri-involutive K3 (TIK3) surface. We explain a
phenomenon noticed by Fuchs, Litman, Silverman, and Tran: over a finite field
of order mod , the points of do not form a single
large orbit under the group generated by the three involutions fixing
two variables and a few other obvious symmetries, but rather admit a partition
into two -invariant subsets of roughly equal size. The phenomenon is
traced to an explicit double cover of the surface.Comment: 4 pages. Accepted at IMR
Harmonic Analysis and Statistics of the First Galois Cohomology Group
We utilize harmonic analytic tools to count the number of elements of the
Galois cohomology group with discriminant-like invariant as . Specifically, Poisson summation produces a
canonical decomposition for the corresponding generating series as a sum of
Euler products for a very general counting problem. This type of decomposition
is exactly what is needed to compute asymptotic growth rates using a Tauberian
theorem. These new techniques allow for the removal of certain obstructions to
known results and answer some outstanding questions on the generalized version
of Malle's conjecture for the first Galois cohomology group.Comment: 18 page
Reflection theorems for number rings
The Ohno-Nakagawa reflection theorem is an unexpectedly simple identity relating the number of \GL_2 \ZZ-classes of binary cubic forms (equivalently, cubic rings) of two different discriminants , ; it generalizes cubic reciprocity and the Scholz reflection theorem. In this paper, we provide a framework for generalizing this theorem using a global and local step. The global step uses Fourier analysis on the adelic cohomology of a finite Galois module, modeled after the celebrated Fourier analysis on used in Tate's thesis. The local step is combinatorial, more elementary but much more mysterious. We establish reflection theorems for binary quadratic forms over number fields of class number , and for cubic and quartic rings over arbitrary number fields, as well as binary quartic forms over \ZZ; the quartic results are conditional on some computational algebraic identities that are probabilistically true. Along the way, we find elegant new results on Igusa zeta functions of conics and the average value of a quadratic character over a box in a local field
Diophantine approximation on conics
Given a conic over , it is natural to ask what real
points on are most difficult to approximate by rational points of
low height. For the analogous problem on the real line (for which the least
approximable number is the golden ratio, by Hurwitz's theorem), the
approximabilities comprise the classically studied Lagrange and Markoff
spectra, but work by Cha-Kim and Cha-Chapman-Gelb-Weiss shows that the spectra
of conics can vary. We provide notions of approximability, Lagrange spectrum,
and Markoff spectrum valid for a general and prove that their
behavior is exhausted by the special family of conics , which has symmetry by the modular group and whose Markoff
spectrum was studied in a different guise by A. Schmidt and Vulakh. The proof
proceeds by using the Gross-Lucianovic bijection to relate a conic to a
quaternionic subring of and
classifying invariant lattices in its -dimensional representation.Comment: 13 pp., incl. 2 table