150 research outputs found
Twists of X(7) and primitive solutions to x^2+y^3=z^7
We find the primitive integer solutions to x^2+y^3=z^7. A nonabelian descent
argument involving the simple group of order 168 reduces the problem to the
determination of the set of rational points on a finite set of twists of the
Klein quartic curve X. To restrict the set of relevant twists, we exploit the
isomorphism between X and the modular curve X(7), and use modularity of
elliptic curves and level lowering. This leaves 10 genus-3 curves, whose
rational points are found by a combination of methods.Comment: 47 page
On the number of cubic orders of bounded discriminant having automorphism group , and related problems
For a binary quadratic form , we consider the action of on
a two-dimensional vector space. This representation yields perhaps the simplest
nontrivial example of a prehomogeneous vector space that is not irreducible,
and of a coregular space whose underlying group is not semisimple. We show that
the nondegenerate integer orbits of this representation are in natural
bijection with orders in cubic fields having a fixed "lattice shape". Moreover,
this correspondence is discriminant-preserving: the value of the invariant
polynomial of an element in this representation agrees with the discriminant of
the corresponding cubic order.
We use this interpretation of the integral orbits to solve three
classical-style counting problems related to cubic orders and fields. First, we
give an asymptotic formula for the number of cubic orders having bounded
discriminant and nontrivial automorphism group. More generally, we give an
asymptotic formula for the number of cubic orders that have bounded
discriminant and any given lattice shape (i.e., reduced trace form, up to
scaling). Via a sieve, we also count cubic fields of bounded discriminant whose
rings of integers have a given lattice shape. We find, in particular, that
among cubic orders (resp. fields) having lattice shape of given discriminant
, the shape is equidistributed in the class group of binary
quadratic forms of discriminant . As a by-product, we also obtain an
asymptotic formula for the number of cubic fields of bounded discriminant
having any given quadratic resolvent field.Comment: 33 page
Averages and moments associated to class numbers of imaginary quadratic fields
For any odd prime , let denote the -part of the
class number of the imaginary quadratic field .
Nontrivial pointwise upper bounds are known only for ; nontrivial
upper bounds for averages of have previously been known only for
. In this paper we prove nontrivial upper bounds for the average of
for all primes , as well as nontrivial upper bounds
for certain higher moments for all primes .Comment: 26 pages; minor edits to exposition and notation, to agree with
published versio
Lifting, restricting and sifting integral points on affine homogeneous varieties
In a previous paper {GN2} an effective solution of the lattice point counting
problem in general domains in semisimple S-algebraic groups and affine
symmetric varieties was established. The method relies on the mean ergodic
theorem for the action of G on G/Gamma, and implies uniformity in counting over
families of lattice subgroups admitting a uniform spectral gap. In the present
paper we extend some methods developed in {NS} and use them to establish
several useful consequences of this property, including : Effective upper
bounds on lifting for solutions of congruences in affine homogeneous varieties,
effective upper bounds on the number of integral points on general subvarieties
of semisimple group varieties, effective lower bounds on the number of almost
prime points on symmetric varieties, and effective upper bounds on almost prime
solutions of Linnik-type congruence problems in homogeneous varieties.Comment: Submitte
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