Hauteurs et mesures de Tamagawa sur les variétés de Fano

Height Zeta Functions of Equivariant Compactifications of Unipotent Groups

“…The main result is the determination of the asymptotic (1.1) for arbitrary biequivariant compactifications X as above and L D K X , the anticanonical line bundle equipped with a smooth adelic metrization, proving Manin's conjecture [9] and its refinement by Peyre [15] for this class of varieties. This generalizes the theorem for equivariant compactifications of the Heisenberg group proved in [18].…”

“…Our main result is a proof of Manin's conjecture: where b D rk Pic.X / D #A is the number of boundary components and . K X / is the Tamagawa number defined by Peyre [15].…”

“…Pic.X / R be the anticanonical class. We know (see Proposition 7.2) that IJ 2 for all 2 A. Conjecturally, analytic properties of height zeta functions Z.sL/ depend on the location of L D .l˛/ 2 Pic.X / with respect to the anticanonical class and the cone ƒ eff .X / (see [1,9,15]). Precisely, define a.L/ WD inffa j aL C K X 2 ƒ eff .X /g D max˛.IJ= l˛/I b.L/ WD #f˛j IJD a.L/l˛g; C.L/ WD f˛j IJ¤ a.L/l˛g; c.L/ WD Q˛… C.L/ l 1 .…”

On Manin's conjecture for a family of Châtelet surfaces

“…For any v ∈ Val(Q), Q v is the corresponding completion of Q. As explained in [Pey95,§2], such a height enables us to define a Tamagawa measure ω H on the adelic space S(A Q ) = v∈Val(Q) S(Q v ). We also consider the constant α(S) defined in [Pey95, Def.…”

Squareful numbers in hyperplanes

“…We refer to [Peyre 1995] for more details on this matter. This implies we can obtain a description of the constant C Q y in terms of the measure ω H y of a certain subset of the adelic space Q y ‫ށ(‬ ‫ޑ‬ ) of the quadric Q y .…”