Variation of the rank in families of elliptic curves

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Elliptic curves have fascinating properties. They can be expressed as the solutions of a cubic equation y^2=x^3+ax+bx, and their set rational points is endowed with a group structure. The rank of this group measures the number of independent elements up to torsion. The main objects of this proposal are 1-parameter families of elliptic curves. I am especially interested in the variation of the rank in dependence of a parameter. The rank of an elliptic curve E is a fundamental although still largely unknown quantity. There is no general method to compute it, and we do not yet know whether the rank is bounded. It is believed that curves with rank as large as 28 are rare, and that half of the elliptic curves should have rank 0, the other half should have rank 1 and the remaining ranks amount to 0% of all elliptic curves. This is referred to as the Goldfeld conjecture. In this research proposal, I plan on studying the distribution of the rank in one-parameter families through two different approaches: analytic and geometric. (1) The analytic approach uses a substitute to the rank: the root number. The Birch and Swinnerton-Dyer conjecture predicts that the root number of E is equal to the parity of the geometric rank. In my recent works, I studied the variation of the root number in both isotrivial and non-isotrivial families of elliptic curves, on rational fibers, and on integer fibers. In the non-isotrivial case, I prove that the two sets of fibers of rational fibers with positive root number (resp. negative) are both infinite, conditional to two analytic number theory conjectures known to hold for polynomials of small degree. The situation changes completely when the parameter varies over the integers as shown by Washington's example of a Z-family of elliptic curves with a constant root number. (2) The geometric approach views a family of elliptic curves as an elliptic surface. This relates to a question raised by Manin in 1974 - is the set of rational points of a rational elliptic surface dense for Zariski topology? Although the Zariski-density of the rational points is known for del Pezzo surfaces of degree 3 and higher, Manin's question is only partially solved in the cases of a del Pezzo surface of degree 1. Moreover, the decreasing of the rank of an elliptic surface by specialization is also a fascinating phenomenon that might be explained through the geometry of del Pezzo surfaces of degree 1. This would lead to a geometric explanation for the decrease in rank which was established by Silverman by other means.