Shimura curves and new abc bounds
Shimura curves and new abc bounds
Existing unconditional progress on the abc conjecture andÌýSzpiro's conjecture is rather limited and coming from essentially only twoÌýapproaches: The theory of linear forms in p-adic logarithms, and bounds forÌýthe degree of modular parametrizations of elliptic curves by usingÌýcongruences of modular forms. In this talk I will discuss a new approach asÌýwell as some unconditional results that it yields. For a fixed ellipticÌýcurve E over the rationals one has several modular parametrizations comingÌýfrom various Shimura curves X, and our method amounts to using ArakelovÌýtheory to bound how these degrees vary as we change the source curve X,Ìýkeeping E fixed. Unlike linear forms in p-adic logarithms, our method isÌýglobal and deals with all local contributions at once. ConcreteÌýunconditional consequences will be discussed, such as bounding the numberÌýof divisors of abc triples polynomially on the radical, bounding theÌýproduct of the ''fudge factors'' of elliptic curves polynomially on theÌýconductor, and new lower bounds for truncated counting functions in theÌýcontext of Vojta's arithmetic conjecture.