1,642 research outputs found
Binary forms as sums of two squares and Ch\^atelet surfaces
The representation of integral binary forms as sums of two squares is
discussed and applied to establish the Manin conjecture for certain Ch\^atelet
surfaces defined over the rationals.Comment: 33 page
Sums of arithmetic functions over values of binary forms
Given a suitable arithmetic function h, we investigate the average order of h
as it ranges over the values taken by an integral binary form F. A general
upper bound is obtained for this quantity, in which the dependence upon the
coefficients of F is made completely explicit.Comment: 12 page
Manin's conjecture for quartic del Pezzo surfaces with a conic fibration
An asymptotic formula is established for the number of rational points of
bounded height on a non-singular quartic del Pezzo surface with a conic bundle
structure.Comment: 61 page
Singular del Pezzo surfaces that are equivariant compactifications
We determine which singular del Pezzo surfaces are equivariant
compactifications of G_a^2, to assist with proofs of Manin's conjecture for
such surfaces. Additionally, we give an example of a singular quartic del Pezzo
surface that is an equivariant compactification of a semidirect product of G_a
and G_m.Comment: 14 pages, main result extended to non-closed ground field
Binary linear forms as sums of two squares
We revisit recent work of Heath-Brown on the average order of the quantity
r(L_1)r(L_2)r(L_3)r(L_4), for suitable binary linear forms L_1,..., L_4, for
integers ranging over quite general regions. In addition to improving the error
term in Heath-Brown's estimate we generalise his result quite extensively.Comment: 34 page
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