Ken Ribet

Title: "Congruences between modular forms and degrees of modular parametrizations"

I will discuss the conjecture of Amod Agashe and William Stein that was the subject of my lecture at the Raynaud conference in Orsay this past June: Each weight-two newform with integer coefficients defines apair of positive integers -- the degree of the modular parametrization of the associated elliptic curve, and a modulus that measures congruences between the given newform and other forms in the same space. The first number is easily seen to divide the second, but the exact relation between the two has remained somewhat mysterious. I shall explain why the quotient of the two numbers is divisible only by primes whose squares divide the level of the newform. This divisibility result gives an affirmative answer to a question of Agashe and Stein; see http://modular.fas.harvard.edu/Tables/degphi_table/index.html for numerical data and related comments and analysis.