On the greatest prime factor of polynomial values and subexponential Szpiro in families

Combining a modular approach to the $abc$ conjecture developed by the second author with the classical method of linear forms in logarithms, we obtain improved unconditional bounds for two classical problems. First, for Szpiro's conjecture when the relevant elliptic curves are members of a one-parameter family (an elliptic surface). And secondly, for the problem of giving lower bounds for the greatest prime factor of polynomial values, in the case of quadratic and cubic polynomials. The latter extends earlier work by the second author for the polynomial $n^2+1$.