Density of rational points on certain surfaces
نویسندگان
چکیده
منابع مشابه
Rational Points on Certain Elliptic Surfaces
Let Ef : y 2 = x + f(t)x, where f ∈ Q[t] \ Q, and let us assume that deg f ≤ 4. In this paper we prove that if deg f ≤ 3, then there exists a rational base change t 7→ φ(t) such that there is a non-torsion section on the surface Ef◦φ. A similar theorem is valid in case when deg f = 4 and there exists t0 ∈ Q such that infinitely many rational points lie on the curve Et0 : y 2 = x + f(t0)x. In pa...
متن کاملRational Points on Certain Del Pezzo Surfaces of Degree One
Let f(z) = z +az + bz + cz+ d ∈ Z[z] and let us consider a del Pezzo surface of degree one given by the equation Ef : x 2 − y − f(z) = 0. In this note we prove that if the set of rational points on the curve Ea, b : Y 2 = X + 135(2a − 15)X − 1350(5a + 2b − 26) is infinite, then the set of rational points on the surface Ef is dense in the Zariski topology.
متن کاملDensity of Rational Points on Diagonal Quartic Surfaces
Let a, b, c, d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in P defined by ax + by + cz + dw = 0. We prove that if V contains a rational point that does not lie on any of the 48 lines on V or on any of the coordinate planes, then the set of rational points on V is dense in both the Zariski topology and the real analytic
متن کاملThe Density of Rational Points on Curves and Surfaces
Let n ≥ 3 be an integer and let F (x) = F (x1, . . . , xn) ∈ Z[x1, . . . , xn] be an absolutely irreducible form of degree d, producing a hypersurface of dimension n − 2 in Pn−1. This paper is primarily concerned with the number of rational points on this hypersurface, of height at most B, say. In order to describe such points we choose representatives x = (x1, . . . , xn) ∈ Z with the xi not a...
متن کاملRational Points on Cubic Surfaces
Let k be an algebraic number eld and F (x0; x1; x2; x3) a non{singular cubic form with coeecients in k. Suppose that the pro-jective cubic k{surface X P 3 k given by F = 0 contains three coplanar lines deened over k, and let U (k) be the set of k{points on X which does not lie on any line on X. We show that the number of points in U (k), with height at most B, is OF;"(B 4=3+") for any " > 0.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Algebra & Number Theory
سال: 2013
ISSN: 1944-7833,1937-0652
DOI: 10.2140/ant.2013.7.835