Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: eess
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
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The categories of this paper are: bayes-an
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat