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NEAROPTIMAL MEAN VALUE ESTIMATES FOR MULTIDIMENSIONAL WEYL SUMS
"... Abstract. We obtain sharp estimates for multidimensional generalisations of Vinogradov’s mean value theorem for arbitrary translationdilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds ..."
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Abstract. We obtain sharp estimates for multidimensional generalisations of Vinogradov’s mean value theorem for arbitrary translationdilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed. 1.
Balanced line bundles and equivariant compactifications of homogeneous spaces
, 2011
"... Abstract. Manin’s conjecture predicts an asymptotic formula for the number of rational points of bounded height on a smooth projective variety X in terms of global geometric invariants of X. The strongest form of the conjecture implies certain inequalities among geometric invariants of X and of its ..."
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Abstract. Manin’s conjecture predicts an asymptotic formula for the number of rational points of bounded height on a smooth projective variety X in terms of global geometric invariants of X. The strongest form of the conjecture implies certain inequalities among geometric invariants of X and of its subvarieties. We provide a general geometric framework explaining these phenomena, via the notion of balanced line bundles, and prove the required inequalities for a large class of equivariant compactifications of homogeneous spaces. 1.
MULTIPLE MIXING FOR ADELE GROUPS AND RATIONAL POINTS
"... Abstract. We prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of H\G, where H is a connected simple algebraic group embedded diagonally into G: = H n. ..."
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Abstract. We prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of H\G, where H is a connected simple algebraic group embedded diagonally into G: = H n.
Height zeta functions of equivariant compactifications of semidirect products of algebraic groups
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ON THE DENSITY OF INTEGER POINTS ON THE GENERALISED MARKOFFHURWITZ AND DWORK
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