Square-free values of a polynomial (and the analytic rank of an elliptic curve)

Let f be a primitive separable polynomial of positive degree in Z[x]. It is conjectured that a positive proportion of the
values are square-free. This is known as the square-free conjecture. In 1953, Erdös showed that for cubics there are
infinitely many square-free values, and in 1967, Hooley gave the result about positive density when $\deg=3$. Although
we know that is suffices to check the conjecture for the factors, it is still not know for irreducible polynomials of degree
greater than $3$. In $h$ variables, Greaves proves the conjecture for degree $\leq 3h$.
In  this talk, we relate this problem to its main motivation: the study of the variation of the analytic rank in a family
of elliptic curves.