  We develop a general framework to study Szpiro's conjecture and the $abc$
conjecture by means of Shimura curves and their maps to elliptic curves,
introducing new techniques that allow us to obtain several unconditional
results for these conjectures. We first prove various general results about
modular and Shimura curves, including bounds for the Manin constant in the case
of additive reduction, a detailed study of maps from Shimura curves to elliptic
curves and comparisons between their degrees, and lower bounds for the
Petersson norm of integral modular forms on Shimura curves. Our main
applications for Szpiro's conjecture and the $abc$ conjecture include improved
effective bounds for the Faltings height of elliptic curves over $\mathbb{Q}$
in terms of the conductor, bounds for products of $p$-adic valuations of the
discriminant of elliptic curves over $\mathbb{Q}$ which are polynomial on the
conductor, and results that yield a modular approach to Szpiro's conjecture
over totally real number fields with the expected dependence on the
discriminant of the field. These applications lie beyond the scope of previous
techniques in the subject. A main difficulty in the theory is the lack of
$q$-expansions, which we overcome by making essential use of suitable integral
models and CM points. Our proofs require a number of tools from Arakelov
geometry, analytic number theory, Galois representations, complex-analytic
estimates on Shimura curves, automorphic forms, known cases of the Colmez
conjecture, and results on generalized Fermat equations.
