The Manin–Peyre formula for a certain biprojective threefold

Abstract

The conjectures of Manin and Peyre are confirmed for a certain threefold.

The Power‐saving Manin–peyre Conjecture for a Senary Cubic

“…where ρ 0 = 0 and ρ q ′ = 1 if q ′ 1. As in lemma 2.7 of [9] we observe that the terms with q ′ 2 do not contribute.…”

“…The work of [9] shows that S1·I 144 coincides with Peyre's prediction for this variety and so Theorem 1 gives Conjecture 1.2 for W 3 .…”

“…It follows that, in the case i = 1, Finally, we notice that after using the Gamma identity cos( πs 2 )Γ(s) = π 1/2 2 s−1 Γ( s 2 )/Γ( 1−s 2 ), which follows from the reflection and duplication formulae for the Gamma function, the integral on the right hand side of (5.20) reduces to the one computed in Lemma 2.10 of [9], where it was shown to be equal to π 2 + 24 log 2 − 3 by means of a long calculation. One could also give a shorter proof of this identity (still requiring some computations) by writing (α 1 (α 1 − 1)α 2 (α 2 − 1)) −1 in terms of its (1-variable) Mellin transform, applying (5.21) and evaluating the resulting integrals.…”

“…In this case, the varieties under consideration are not equivariant compactifications, the rank of the Picard group of W n is 2 n − n and the subset where x 1 · · · x n y 1 · · · y n = 0 is an accumulating subset. In recent work [9], Blomer, Brüdern and Salberger showed that Conjecture 1.1 holds for W 3 using Fourier analysis. Using recent results of the first author [4], we are able to refine the aforementioned result [9] proving the stronger Conjecture 1.2 for W 3 .…”

“…In recent work [9], Blomer, Brüdern and Salberger showed that Conjecture 1.1 holds for W 3 using Fourier analysis. Using recent results of the first author [4], we are able to refine the aforementioned result [9] proving the stronger Conjecture 1.2 for W 3 . The leading coefficient of P 1 is equal to S1·I 144 , where and χ X denotes the characteristic function of a set X.…”