Lang to Langlands
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Lang — “Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century.” — Serge Lang, preface to Elliptic Functions (second edition, 1987) Langlands — “The theory of modular functions and modular forms, defined on the upper half-plane H and subject to appropriate tranformation laws with respect to the group Gamma = SL(2, Z) of fractional linear transformations, is closely related to the theory of elliptic curves, because the family of all isomorphism classes of elliptic curves over C can be parametrized by the quotient Gamma\H. This is an important, although formal, relation that assures that this and related quotients have a natural structure as algebraic curves X over Q. The relation between these curves and elliptic curves predicted by the Taniyama-Weil conjecture is, on the other hand, far from formal.” — Robert P. Langlands, review of Elliptic Curves , by Anthony W. Knapp. (The review appeared in Bulletin of the American Mathematical Society , January 1994.) |