Improvements in Birch’s theorem on forms in many variables

Abstract

Abstract. We show that a non-singular integral form of degree d is soluble over the integers if and only if it is soluble over R and over Q p for all primes p, provided that the form has at least (d −

On the Hasse principle for quartic hypersurfaces

“…A main difference here from Lemma 3.3 is that the sum (3.12) is a purely cubic exponential sum only containing the form F h , as opposed to a mixture of cubic and quartic exponential sums (3.5), appearing in Lemma 3.3. In Section 9, we will apply Weyl bound directly to estimate S h (α), via [3,Lemma 3.3]. This bound will turn out to be important for us.…”

“…Hanselmann [9] then established the case n = 40. The methodology in [2] has since been generalised by Browning and Prendiville [3], thus improving upon Birch's bounds for every degree d 5. In the special case of diagonal forms F = a 1 x 4 1 + .…”

Counting rational points on hypersurfaces and higher order expansions

“…We first recall some conventions from [3]. We say that a pair α ∈ R/Z and q ∈ N is primitive, if there is some r ∈ Z with (r, q) = 1 and qα = |qα − r|.…”

“…Note that we do not need an explicit dependence on the bound for S ω (α, P ) depending on the coefficients of F (x) which can be found in the formulation of Lemma 3.3 in [3].…”

“…These results include for example improvements in the cubic case due to Heath-Brown [7], [5], a new version of the circle method by Heath-Brown in [6], improvements in the quartic case by Browning and HeathBrown [2] and improvements on the bound (1.1) by Browning and Prendiville [3].…”