Counting rational points on quartic del Pezzo surfaces with a rational conic

Abstract

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over Q that contains a conic defined over Q.

Rational points of bounded height on general conic bundle surfaces

“…These results were obtained in order to assist with future proofs of Manin's conjecture for del Pezzo surfaces with a conic bundle, and to help with applications of Theorem . The equations obtained here, together with the analytic tools in this paper, have already found applications to Manin's conjecture for quartic del Pezzo surfaces .…”

“…Since ω v is continuous on U 0 by Lemma 4.3, it has a minimum in D v , which is non-zero thanks to (5). This shows that ω v (s, t) Dv T −2mv for (s, t) ∈ T D v , which proves (6).…”

“…Our approach yields several new results that were out of reach of previous methods, even over Q. It has already found further applications to Manin's conjecture for quartic del Pezzo surfaces in the recent paper [6].…”

“…Moreover, since Δ(s 0 , t 0 ) = 0, we may achieve (3) and (4) by further shrinking the balls D v . Property (5) is clear from Lemma 4.2, so it remains to prove (6). To this end, we use the homogeneity condition (4.3) on Q(s, t, x 0 , x 1 , x 2 ) and the shape (4.2) of the local height factors H (s,t),v = H v • φ (s,t) at archimedean places.…”

“…During the conference ‘Higher dimensional varieties and rational points’ at Budapest in 2001, Salberger has announced a proof of for any , whenever contains a conic over , and this work was subsequently extended to number fields by Browning and Swarbrick‐Jones [, Theorem 1.1]. Even more is known now for due to very recent work of Browning and Sofos , building in part upon the results of the present paper.…”