1,037 research outputs found
Rings and ideals parametrized by binary n-ic forms
The association of algebraic objects to forms has had many important
applications in number theory. Gauss, over two centuries ago, studied quadratic
rings and ideals associated to binary quadratic forms, and found that ideal
classes of quadratic rings are exactly parametrized by equivalence classes of
integral binary quadratic forms. Delone and Faddeev, in 1940, showed that cubic
rings are parametrized by equivalence classes of integral binary cubic forms.
Birch, Merriman, Nakagawa, Corso, Dvornicich, and Simon have all studied rings
associated to binary forms of degree n for any n, but it has not previously
been known which rings, and with what additional structure, are associated to
binary forms. In this paper, we show exactly what algebraic structures are
parametrized by binary n-ic forms, for all n. The algebraic data associated to
an integral binary n-ic form includes a ring isomorphic to as a
-module, an ideal class for that ring, and a condition on the ring
and ideal class that comes naturally from geometry. In fact, we prove these
parametrizations when any base scheme replaces the integers, and show that the
correspondences between forms and the algebraic data are functorial in the base
scheme. We give geometric constructions of the rings and ideals from the forms
that parametrize them and a simple construction of the form from an appropriate
ring and ideal.Comment: submitte
Universality and the circular law for sparse random matrices
The universality phenomenon asserts that the distribution of the eigenvalues
of random matrix with i.i.d. zero mean, unit variance entries does not depend
on the underlying structure of the random entries. For example, a plot of the
eigenvalues of a random sign matrix, where each entry is +1 or -1 with equal
probability, looks the same as an analogous plot of the eigenvalues of a random
matrix where each entry is complex Gaussian with zero mean and unit variance.
In the current paper, we prove a universality result for sparse random n by n
matrices where each entry is nonzero with probability where
is any constant. One consequence of the sparse universality
principle is that the circular law holds for sparse random matrices so long as
the entries have zero mean and unit variance, which is the most general result
for sparse random matrices to date.Comment: Published in at http://dx.doi.org/10.1214/11-AAP789 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Parametrizing quartic algebras over an arbitrary base
We parametrize quartic commutative algebras over any base ring or scheme
(equivalently finite, flat degree four -schemes), with their cubic
resolvents, by pairs of ternary quadratic forms over the base. This generalizes
Bhargava's parametrization of quartic rings with their cubic resolvent rings
over by pairs of integral ternary quadratic forms, as well as
Casnati and Ekedahl's construction of Gorenstein quartic covers by certain rank
2 families of ternary quadratic forms. We give a geometric construction of a
quartic algebra from any pair of ternary quadratic forms, and prove this
construction commutes with base change and also agrees with Bhargava's explicit
construction over .Comment: submitte
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