The research areas of the LMFBD are distinguished by
the variety of connections among the various objects.

Algebraic Varieties an automasdkldaskla Automorphic Forms L-functions Number Fields Artin Representations     component             component Galois Groups The Rankin-Selberg convolution of a degree d1 L-function and a degree d2 L-function is an L-function of degree d1d2.   Rankin-Selberg   Galois   closure Smooth varieties defined over a number field field of definition A "lift" is a map from the automorphic forms on one group to the automorphic forms on another group.         lifts A "twist" is ...          twists The Langlands program predicts that Artin representations are automorphic. give explicit examples of Dedekind zeta function The local factors of the Hasse-Weil L-function are determined by counting points on the variety modulo prime powers. Hasse-Weil L-function 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 $L(s,\pi,\rho)$. automorphic representation A variety is 'modular' if its Hasse-Weil L-function is the same as the standard L-function of some modular form. The key step in proving Fermat's Last Theorem was showing that certain elliptic curves are modular.     some varieties are modular Say something about Shimura varieties.             Shimura varieties of the rational numbers. Associated to any number field is a Galois group: the Galois group of its Galois closure The local factors of the Artin L-function are constructed from the trace of Frobenius.