Rational points on certain quintic hypersurfaces (original) (raw)

Rational points on certain elliptic surfaces

Acta Arithmetica, 2007

\ 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 → ϕ(t) such that there is a non-torsion section on the surface E f •ϕ . A similar theorem is valid in case when deg f = 4 and there exists t 0 ∈ Q such that infinitely many rational points lie on the curve Et 0 : y 2 = x 3 + f (t 0 )x. In particular, we prove that if deg f = 4 and f is not an even polynomial, then there is a rational point on E f . Next, we consider a surface E g : y 2 = x 3 + g(t), where g ∈ Q[t] is a monic polynomial of degree six. We prove that if the polynomial g is not even, there is a rational base change t → ψ(t) such that on the surface E g•ψ there is a non-torsion section. Furthermore, if there exists t 0 ∈ Q such that on the curve E t 0 : y 2 = x 3 +g(t 0 ) there are infinitely many rational points, then the set of these t 0 is infinite. We also present some results concerning diophantine equation of the form x 2 − y 3 − g(z) = t, where t is a variable.

Quartic surface, its bitangents and rational points

arXiv: Number Theory, 2020

Let X be a smooth quartic surface not containing lines, defined over a number field K. We prove that there are only finitely many bitangents to X which are defined over K. This result can be interpreted as saying that a certain surface, having vanishing irregularity, contains only finitely many rational points. In our proof, we use the geometry of lines of the quartic double solid associated to X. In a somewhat opposite direction, we show that on any quartic surface X over a number field K, the set of algebraic points in X(\overeline K) which are quadratic over a suitable finite extension K' of K is Zariski-dense.

Rational Points on Certain Hyperelliptic Curves over Finite Fields

Bulletin of the Polish Academy of Sciences Mathematics, 2007

Let K be a field, a, b ∈ K and ab = 0. Let us consider the polynomials g1(x) = x n + ax + b, g2(x) = x n + ax 2 + bx, where n is a fixed positive integer. In this paper we show that for each k ≥ 2 the hypersurface given by the equation

On surfaces of general type with q = 5

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2012

We prove that a complex surface S with irregularity q(S) = 5 that has no irrational pencil of genus > 1 has geometric genus pg(S) ≥ 8. As a consequence, one is able to classify minimal surfaces S of general type with q(S) = 5 and pg(S) < 8. This result is a negative answer, for q = 5, to the question asked in [MP1] of the existence of surfaces of general type with irregularity q ≥ 4 that have no irrational pencil of genus > 1 and with the lowest possible geometric genus pg = 2q−3. This gives some evidence for the conjecture that the only irregular surface with no irrational pencil of genus > 1 and pg = 2q − 3 is the symmetric product of a genus three curve.

4 Constructions of Diagonal Quartic and Sextic Surfaces with Infinitely Many Rational Points

2016

In this note we construct several infinite families of diagonal quartic surfaces ax 4 + by 4 + cz 4 + dw 4 = 0, where a, b, c, d ∈ Z \ {0} with infinitely many rational points and satisfying the condition abcd =. In particular, we present an infinite family of diagonal quartic surfaces defined over Q with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type ax 6 + by 6 + cz 6 + dw i = 0, i = 2, 3, or 6, with infinitely many rational points.

Covering Techniques and Rational Points on Some Genus 5 Curves

Contemporary Mathematics, 2015

We describe a method that allows, under some hypotheses, computation of all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.

On representing coordinates of points on elliptic curves by quadratic forms

Acta Arithmetica

Given an elliptic quartic of type Y 2 = f (X) representing an elliptic curve of positive rank over Q, we investigate the question of when the Y-coordinate can be represented by a quadratic form of type ap 2 + bq 2. In particular, we give examples of equations of surfaces of type c 0 + c 1 x + c 2 x 2 + c 3 x 3 + c 4 x 4 = (ap 2 + bq 2) 2 , a, b, c ∈ Q where we can deduce the existence of infinitely many rational points. We also investigate surfaces of type Y 2 = f (ap 2 + bq 2) where the polynomial f is of degree 3.