The largest prime factor of $n^2+1$ and improvements on subexponential $ABC$

We combine transcendental methods and the modular approaches to the $ABC$ conjecture to show that the largest prime factor of $n^2+1$ is at least of size $(\log_2 n)^2/\log_3n$ where $\log_k$ is the $k$-th iterate of the logarithm. This gives a substantial improvement on the best available estimates, which are essentially of size $\log_2 n$ going back to work of Chowla in 1934. Using the same ideas, we also obtain significant progress on subexpoential bounds for the $ABC$ conjecture, which in a case gives the first improvement on a result by Stewart and Yu dating back over two decades. Central to our approach is the connection between Shimura curves and the $ABC$ conjecture developed by the author.