On the density of rational points on rational elliptic surfaces

Abstract

Let E → P 1 Q be a non-trivial rational elliptic surface over Q with base P 1 Q (with a section). We conjecture that any non-trivial elliptic surface has a Zariski-dense set of Q-rational points. In this paper we work on solving the conjecture in case E is rational by means of geometric and analytic methods. First, we show that for E rational, the set E (Q) is Zariski-dense when E is isotrivial with non-zero j-invariant and when E is non-isotrivial with a fiber of type II * , III * , IV * or I * m (m ≥ 0). We also use the parity conjecture to prove analytically the density on a certain family of isotrivial rational elliptic surfaces with j = 0, and specify cases for which neither of our methods leads to the proof of our conjecture.

Geometry of the del Pezzo surface y 2 =x 3 +Am 6 +Bn 6

“…Suppose that it is not the case. Then E has generic rank r E = 1 if there exist some non-zero p, q ∈ Z coprime and not divisible by 2 and 3, such that A and B, written up to sixth power representative and possibly switched in the equation (4) , are among…”

“…Building on ideas of [10,15] and [27], the first author [5] confirms (conditionally (3) ) that non-isotrivial elliptic surfaces have a dense set of rational point through the study of the variation of the root number. For isotrivial elliptic surfaces, it can happen that a family has a constant root number, but [4] proves (unconditionally) that they have a dense set of rational points given that the surface has j-invariant j ̸ = 0. If W (E t ) = +1 for all t ∈ P 1 (Q) and that the j-invariant is 0, then it is not yet known whether the rational points are Zariski dense.…”

“…Hence, for each i we have set of sextic twists for which it is not guaranteed that the root number of a fibre E t varies when t varies through t ∈ P(Q). In a previous paper [4,Theorem 6.1], the first author gives the precise conditions on a, b and c for which the root number takes the same values on every fibre.…”

Ranks of elliptic curves over $\mathbb{Q}(T)$ of small degree in $T$

“…For example, Silverman's specialization theorem asserts that for almost all specializations of T at t ∈ Q, the rank of the associated elliptic curve defined over Q is at least r E : this has direct consequences on the study of the distribution of ranks of families of elliptic curves defined over Q, or over number fields K [Mil06, DHP15, ST95, RS01] and on the research of elliptic curves with high rank [Mes91,Fer97,ALRM07]. The study of the rank, r E , has also some impact on a specific question in arithmetic geometry, concerning whether the set E(Q) is Zariski-dense in E: the question is positively answered whenever r E > 0, and if r E = 0 a sufficient criterion consists in showing that there exist infinitely many specializations of T at t ∈ Q such that the rank of the corresponding curve over Q is positive (see [Maz92] and recent works by J. Desjardins on the subject, [Des19,Des18]). For example, under the parity conjecture this can be done by studying the behavior of the root numbers of the specializations.…”

On the typical rank of elliptic curves over ${\mathbb Q}(T)$

“…This has direct consequences on the study of the distribution of ranks of families of elliptic curves defined over Q, or over number fields K [Mil06, DHP15, ST95, RS01] and on the research of high rank elliptic curves [Mes91,Fer97,ALRM07]. The study of the rank, r E , has also some impact on a question in arithmetic geometry asking whether the set E(Q) is Zariski-dense in E. The question is positively answered whenever r E > 0, and if r E = 0 a sufficient criterion consists in showing that there exist infinitely many specializations of T at t ∈ Q such that the rank of the corresponding curve over Q is positive (see [Maz92] and recent works by J. Desjardins on the subject, [Des19a,Des18]). Under the parity conjecture this can be done, for example, by studying the behavior of the root numbers of the specializations.…”