We prove that there exists an integer p0 such that X split (p)(Q) is made of cusps and CM-points for any prime p > p0. Equivalently, for any non-CM elliptic curve E over Q and any prime p > p0 the… (More)

It is well-known since Gauss that infinitely many quadratic fields have even class number. In fact, if K is a quadratic field of discriminant D, having r prime divisors, then the class number hK is… (More)

Given a polynomial with integer coefficients, we calculate the density of the set of primes modulo which the polynomial has a root. We also give a simple criterion to decide whether or not the… (More)

We bound the j-invariant of integral points on a modular curve in terms of the congruence group defining the curve. We apply this to prove that the modular curve Xsplit(p ) has no non-trivial… (More)

1. Introduction. The history of numerical solution of Diophantine equations began in 1969, when Baker and Davenport [1] solved completely a system of two Pell equations. They used the well-known fact… (More)

We bound the j-invariant of S-integral points on arbitrary modular curves over arbitrary fields, in terms of the congruence group defining the curve, assuming a certain Runge condition is satisfied… (More)

We prove that integral points can be effectively determined on all but finitely many modular curves, and on all but one modular curve of prime power level.

Theorem 1.1 (Chevalley-Weil) Let Ṽ φ → V be a finite étale covering of normal projective varieties, defined over a number field K. Then there exists a non-zero integer T such that for any P ∈ V (K)… (More)