17,069 research outputs found
Most hyperelliptic curves over Q have no rational points
By a hyperelliptic curve over Q, we mean a smooth, geometrically irreducible,
complete curve C over Q equipped with a fixed map of degree 2 to P^1 defined
over Q. Thus any hyperelliptic curve C over Q of genus g can be embedded in
weighted projective space P(1,1,g+1) via an equation of the form
C : z^2 = f(x,y) = f_0 x^n + f_1 x^{n-1} y + ... + f_n y^n where n=2g+2, the
coefficients f_i lie in Z, and f factors into distinct linear factors over
Q-bar. Define the height H(C) of C by H(C):=max{|f_i|}, and order all
hyperelliptic curves over Q of genus g by height. Then we prove that, as g
tends to infinity:
1) a density approaching 100% of hyperelliptic curves of genus g have no
rational points;
2) a density approaching 100% of those hyperelliptic curves of genus g that
have points everywhere locally fail the Hasse principle; and
3) a density approaching 100% of hyperelliptic curves of genus g have empty
Brauer set, i.e., have a Brauer-Manin obstruction to having a rational point.
We also prove positive proportion results of this type for individual genera,
including g = 1.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1208.100
The density of discriminants of quintic rings and fields
We determine, asymptotically, the number of quintic fields having bounded
discriminant. Specifically, we prove that the asymptotic number of quintic
fields having absolute discriminant at most X is a constant times X. In
contrast with the quartic case, we also show that a density of 100% of quintic
fields, when ordered by absolute discriminant, have Galois closure with full
Galois group . The analogues of these results are also proven for orders
in quintic fields. Finally, we give an interpretation of the various constants
appearing in these theorems in terms of local masses of quintic rings and
fields
Adelic versions of the Weierstrass approximation theorem
Let be a compact subset of
and denote by
the ring of continuous
functions from into . We obtain two kinds
of adelic versions of the Weierstrass approximation theorem. Firstly, we prove
that the ring is dense in the
direct product for the
uniform convergence topology. Secondly, under the hypothesis that, for each
, for all but finitely many , we prove the
existence of regular bases of the -module , and show that, for such
a basis , every function in
may be uniquely written
as a series where
and .Comment: minor corrections the statement of Theorem 3.5, which covers the case
of a general compact subset of the profinite completion of Z. to appear in
Journal of Pure and Applied Algebra, comments are welcome
Globalization, Literacy Levels, and Economic Development
This paper estimated models for GDP growth rates, poverty levels, and inequality measures for the period 1990?2000 using data on 54 developing countries at five-yearly intervals. Issues of globalization were investigated by analysing the differential effects of the countries? exports and imports and by postulating trans-logarithmic models that allow for non-linear effects of literacy levels and measures of openness. The main findings were that literacy rates affected growth rates in a quadratic manner and countries with higher literacy were more likely to benefit from globalization. Second, the model for growth rates showed non-linear and differential effects of the export/GDP and import/GDP ratios. Third, the models indicated that population health indicators such as life expectancy were important predictors of GDP growth rates. Fourth, models for poverty measures showed that poverty was not directly affected by globalization indicators. Finally, the model for Gini coefficients indicated significant effects of ?medium? and ?high? skilled labour work force, with higher proportions of high-skilled labour implying greater inequality.globalization, economic development, education, endogeneity, inequality, poverty, non-linearities, trade
On the mean number of 2-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields
Given any family of cubic fields defined by local conditions at finitely many
primes, we determine the mean number of 2-torsion elements in the class groups
and narrow class groups of these cubic fields when ordered by their absolute
discriminants.
For an order in a cubic field, we study the three groups: , the group of ideal classes of of order 2; , the group of narrow ideal classes of of order 2; and
, the group of ideals of of order 2. We prove that
the mean value of the difference is always equal to , whether one averages over the maximal orders in
real cubic fields, over all orders in real cubic fields, or indeed over any
family of real cubic orders defined by local conditions. For the narrow class
group, we prove that the mean value of the difference is equal to for any such family. For any family
of complex cubic orders defined by local conditions, we prove similarly that
the mean value of the difference is always equal to , independent of the family.
The determination of these mean numbers allows us to prove a number of
further results as by-products. Most notably, we prove---in stark contrast to
the case of quadratic fields---that: 1) a positive proportion of cubic fields
have odd class number; 2) a positive proportion of real cubic fields have
isomorphic 2-torsion in the class group and the narrow class group; and 3) a
positive proportion of real cubic fields contain units of mixed real signature.
We also show that a positive proportion of real cubic fields have narrow class
group strictly larger than the class group, and thus a positive proportion of
real cubic fields do not possess units of every possible real signature.Comment: 17 page
The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields
For , 4, and 5, we prove that, when -number fields of degree
are ordered by their absolute discriminants, the lattice shapes of the rings of
integers in these fields become equidistributed in the space of lattices.Comment: 12 page
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