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Local Fields

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Counting Rational Points on Algebraic Varieties

For any N � 2, let ZP N be a geometrically integral algebraic variety of degree d. This paper is concerned with the number NZ(B) of Q-rational points on Z which have height at most B. For any " > 0…
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Forms in many variables and differing degrees

We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak…
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Quantitative Arithmetic of Projective Varieties

The Manin conjectures.- The dimension growth conjecture.- Uniform bounds for curves and surfaces.- A1 del Pezzo surface of degree 6.- D4 del Pezzo surface of degree 3.- Siegel's lemma and…
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Power-free values of polynomials

For an irreducible polynomial in at most two variables the problem of representing power-free integers is investigated.
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Analytic Methods for Diophantine Equations and Diophantine Inequalities: Cubic forms: bilinear equations

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Rational points on quartic hypersurfaces

Abstract Let X be a projective non-singular quartic hypersurface of dimension 39 or more, which is defined over ℚ. We show that X(ℚ) is non-empty provided that X(ℝ) is non-empty and X has p-adic…
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Averages of shifted convolutions of d3(n)

Abstract We investigate the first and second moments of shifted convolutions of the generalized divisor function d3(n).
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Norm forms for arbitrary number fields as products of linear polynomials

Let K/Q be a field extension of finite degree and let P(t) be a polynomial over Q that splits into linear factors over Q. We show that any smooth model of the affine variety defined by the equation…
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A transference approach to a Roth-type theorem in the squares

We show that any subset of the squares of positive relative upper density contains non-trivial solutions to a translation-invariant linear equation in five or more variables, with explicit…
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