6,169 research outputs found
Many cubic surfaces contain rational points
Building on recent work of Bhargava--Elkies--Schnidman and Kriz--Li, we
produce infinitely many smooth cubic surfaces defined over the field of
rational numbers that contain rational points.Comment: 23 pages; minor edits and added new remark (Remark 2.1) following an
argument of Jahne
The Manin conjecture in dimension 2
These lecture notes describe the current state of affairs for Manin's
conjecture in the context of del Pezzo surfaces.Comment: 57 pages. These are a preliminary version of lecture notes for the
"School and conference on analytic number theory", ICTP, Trieste,
23/04/07-11/05/0
Density of integer solutions to diagonal quadratic forms
Let Q be a non-singular diagonal quadratic form in at least four variables.
We provide upper bounds for the number of integer solutions to the equation
Q=0, which lie in a box with sides of length 2B, as B tends to infinity. The
estimates obtained are completely uniform in the coefficients of the form, and
become sharper as they grow larger in modulus.Comment: 23 page
Equal sums of like polynomials
Let be a polynomial of degree , with integer coefficients. Then the
paucity of non-trivial positive integer solutions to the equation
is established. The corresponding situation for equal
sums of three like polynomials is also investigated.Comment: 8 pages; to appear in Bull. London Math. So
Counting rational points on quartic del Pezzo surfaces with a rational conic
Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over Q that contains a conic defined over Q
Power-free values of polynomials on symmetric varieties
Given a symmetric variety Y defined over the rationals and a non-zero
polynomial with integer coefficients, we use techniques from homogeneous
dynamics to establish conditions under which the polynomial can be made r-free
for a Zariski dense set of integral points on Y. We also establish an
asymptotic counting formula for this set. In the special case that Y is a
quadric hypersurface, we give explicit bounds on the size of r by combining the
argument with a uniform upper bound for the density of integral points on
general affine quadrics.Comment: 47 pages; accepted versio
Varieties with too many rational points
We investigate Fano varieties defined over a number field that contain
subvarieties whose number of rational points of bounded height is comparable to
the total number on the variety.Comment: 23 page
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