Ken Ribet

Abstract: Classical modular forms give rise to families of 2-dimensional l-adic representations of the Galois group of Q. These representations typically have "coefficients in E", where E is a number field. In other words, these representations form a compatible system of lambda-adic representations, where lambda runs over the set of non-Archimedian primes of E. For various applications, one seeks to prove that the images of these representations are large. I shall discuss a tool for studying the image which works even when lambda is a ramified prime of E. The criterion applies in the study of the l-adic representations attached to the abelian variety J=J_0(N), where N is a prime number. Namely, the l-adic representation attached to J has large image provided that: l is bigger than 5; l is not an Eisenstein prime; and l^2 does not divide the discriminant of the endomorphism ring of J.