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

Algebraic Varieties Automorphic Forms L-functions Number Fields Artin Representations Galois Groups The Rankin-Selberg convolution of a degree $d_1$ L-function and a degree $d_2$ L-function in an L-funciton of degree $d_1 d_2$.   Rankin-Selberg   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 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 Hasse-Weil L-function are determined by counting points on the variety modulo prime powers. Hasse-Weil L-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 $L(s,\pi,\rho)$. 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 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.)