Modular forms and calabi-yau varieties

Abstract

Let be a holomorphic newform of weight k ≥ 2 relative to Γ(N) acting on the upper half plane H. Suppose the coefficients an are all rational. When k = 2, a celebrated theorem of Shimura asserts that there corresponds an elliptic curve E over Q such that for all primes. Equivalently, there is, for every prime l, an l-adic representation ρl of the absolute Galois group of Q, given by its action on the l-adic Tate module of E, such that ap is, for any, the trace of the Frobenius Frp at p on ρl