92 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
A positive proportion of Thue equations fail the integral Hasse principle
For any nonzero , we prove that a positive proportion of
integral binary cubic forms do locally everywhere represent but do not
globally represent ; that is, a positive proportion of cubic Thue equations
fail the integral Hasse principle. Here, we order all classes of
such integral binary cubic forms by their absolute discriminants. We prove
the same result for Thue equations of any fixed degree ,
provided that these integral binary -ic forms are ordered by the maximum
of the absolute values of their coefficients.Comment: Previously cited as "A positive proportion of locally soluble Thue
equations are globally insoluble", Two typos are fixed and small mathematical
error in Section 4 is correcte
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
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