Algebraic Varieties Hyperelliptic Curves Jacobian Elliptic Curves weight 2 cusp forms on GL(2) Abelian Surfaces weight 2 cusp forms on the paramodular group K(N)   ask Kiran Kedlaya C-Y 3-folds     weight 4 cusp forms on GL(2) K3 Surfaces     weight 3 cusp forms on GL(2) Calabi-Yau Manifolds Abelian Varieties   A "lift" is a map from the automorphic forms on one group to the automorphic forms on another group.         lifts The Langlands program predicts that Artin representations are automorphic. give explicit examples of The Dedekind zeta function is defined as a product over the prime ideals of the number field Dedekind zeta function The local factors of the Artin L-function are constructed from the trace of Frobenius. Artin L-function A "comnverse theorem" is an assertion that an L-function is the L-function of an automorphic object. converse theorem Given a cuspidal automorphic representation $\pi$ of a XXXX group $G$ and a finite dimensional representation $\rho$ of the Weil-Deligne group of $G$, one can form the automorphic L-function $L(s,\pi,\rho)$. representations of the Weil-Deligne group An Artin representation is a finite dimensional representation of the Galois group of a number field. (Somebody please correct that to what it should say.)