Abstract: A classical technique to upper bound ranks of elliptic curve is descent. The case of 2-descent, when studied for quadratic twist families of elliptic curves has attracted the attention of several authors, Heath-Brown, Friedlander--Iwaniec--Mazur--Rubin, Kane, Smith, among some of them, giving statistical result on the behavior of 2-Selmer in natural families. Motivated by Silverman's conjecture (a version of Goldfeld conjecture for general polynomial families of elliptic curves) and by questions in mathematical logic, such as Hilbert 10th problem (where one wants to upper bound 2-Selmer on a polynomial family of curves having positive rank), there has been recent interest in studying these problems in significantly thinner families than usual, such as those given by values of polynomials. I will overview the methods used and some of the recent results obtained for such thin families. This is joint work with Peter Koymans.