Rational points on diagonal quartic surfaces

Abstract

Abstract. We searched up to height 10 7 for rational points on diagonal quartic surfaces. The computations fill several gaps in earlier lists computed by Pinch, Swinnerton-Dyer, and Bright.

Constructions of diagonal quartic and sextic surfaces with infinitely many rational points

“…This latter point is of infinite order, and hence its multiples pull back to infinitely many rational points on (8). Using the same type of reasoning as in the proof of Theorem 2.1, we get the following.…”

“…where X i = ϕ i (p, q), i = 1, 2, 3, is a parametric solution of the equation X 2 1 + 3X 2 2 = kX 2 3 . We also note that some believe a small Picard number of S has a strong influence on the set of rational points on S. This belief was questioned in a recent paper of Elsenhans [8], who considered the set of surfaces ax 4 + by 4 = cz 4 + dw 4 with 1 ≤ a, b, c, d ≤ 15 from a computational point of view. He noted that in the considered range there is no great difference between the number of unsolved cases for differing Picard rank.…”

“…On setting z = (p 2 + pq + q 2 )rZ/(p + q), w = (p 2 + pq + q 2 )W , the surface now takes the form (8) V :…”

“…By reducing modulo a large number of primes, we believe the curve to be absolutely irreducible. A small search produces zeros at (a, c) = (0, 0), (1, 0), (2, 0), (1/2, 0), (2/9, 0), (0, 2), (0, 3), (0, −1/23), (1, 2), (2, 3), (2,4), (1, 3/2), (1, 3/2), (−2, 1/3), (3/2, 3), (8,3), (1, 7/3), but no new identities result. Case-by-case analysis now results in the following table.…”