380 research outputs found
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
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
-Frequent hypercyclicity in spaces of operators
We provide conditions for a linear map of the form to be
-frequently hypercyclic on algebras of operators on separable Banach spaces.
In particular, if is a bounded operator satisfying the -Frequent
Hypercyclicity Criterion, then the map = is shown to be
-frequently hypercyclic on the space of all compact
operators and the real topological vector space of all
self-adjoint operators on a separable Hilbert space . Further we provide a
condition for to be -frequently hypercyclic on the Schatten von
Neumann classes . We also characterize frequent hypercyclicity of
on the trace-class of the Hardy space, where the
symbol denotes the multiplication operator associated to .Comment: The previous version has been changed considerably with many
corrections rectifie
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
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