221 research outputs found
Heuristics for the Brauer-Manin obstruction for curves
We conjecture that if C is a curve of genus >1 over a number field k such
that C(k) is empty, then a method of Scharaschkin (equivalent to the
Brauer-Manin obstruction in the context of curves) supplies a proof that C(k)
is empty. As evidence, we prove a corresponding statement in which C(F_v) is
replaced by a random subset of the same size in J(F_v) for each residue field
F_v at a place v of good reduction for C, and the orders of Jacobians over
finite fields are assumed to be smooth (in the sense of having only small prime
divisors) as often as random integers of the same size. If our conjecture
holds, and if Shafarevich-Tate groups are finite, then there exists an
algorithm to decide whether a curve over k has a k-point, and the Brauer-Manin
obstruction to the Hasse principle for curves over the number fields is the
only one.Comment: 7 page
Multivariable polynomial injections on rational numbers
For each number field k, the Bombieri-Lang conjecture for k-rational points
on surfaces of general type implies the existence of a polynomial f(x,y) in
k[x,y] inducing an injection k x k --> k.Comment: 4 page
Smooth hypersurface sections containing a given subscheme over a finite field
We use the "closed point sieve" to prove a variant of a Bertini theorem over
finite fields. Specifically, given a smooth quasi-projective subscheme X of P^n
of dimension m over F_q, and a closed subscheme Z in P^n such that Z intersect
X is smooth of dimension l, we compute the fraction of homogeneous polynomials
vanishing on Z that cut out a smooth subvariety of X. The fraction is positive
if m>2l.Comment: 7 pages. This paper appeared a few years ago. (I'm posting it in
response to a request for the TeX file.
Characterizing integers among rational numbers with a universal-existential formula
We prove that Z in definable in Q by a formula with 2 universal quantifiers
followed by 7 existential quantifiers. It follows that there is no algorithm
for deciding, given an algebraic family of Q-morphisms, whether there exists
one that is surjective on rational points. We also give a formula, again with
universal quantifiers followed by existential quantifiers, that in any number
field defines the ring of integers.Comment: 6 page
The Grothendieck ring of varieties is not a domain
Let k be a field. Let K_0(V_k) denote the quotient of the free abelian group
generated by the geometrically reduced varieties over k, modulo the relations
of the form [X]=[X-Y]+[Y] whenever Y is a closed subvariety of X. Product of
varieties makes K_0(V_k) into a ring. We prove that if the characteristic of k
is zero, then K_0(V_k) is not a domain.Comment: 4 page
Squarefree values of multivariable polynomials
Given f in Z[x_1,...,x_n], we compute the density of x in Z^n such that f(x)
is squarefree, assuming the abc conjecture. Given f,g in Z[x_1,...,x_n], we
compute unconditionally the density of x in Z^n such that gcd(f(x),g(x))=1.
Function field analogues of both results are proved unconditionally. Finally,
assuming the abc conjecture, given f in Z[x], we estimate the size of the image
of f({1,2,...,n}) in (Q^*/Q^*2) union {0}.Comment: 16 pages, Latex 2e, will appear in Duke Mathematical Journa
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