Recent Developments in Relation With Modular Theory
A well-known example is the Taniyama–Shimura conjecture, now the modularity theorem, which proposed that each elliptic curve over the rational numbers can be translated into a modular form (in such a way as to preserve the associated L-function). There are difficulties in identifying this with an isomorphism, in any strict sense of the word. Certain curves had been known to be both elliptic curves (of genus 1) and modular curves, before the conjecture was formulated (about 1955). The surprising part of the conjecture was the extension to factors of Jacobians of modular curves of genus > 1. It had probably not seemed plausible that there would be 'enough' such rational factors, before the conjecture was enunciated; and in fact the numerical evidence was slight until around 1970, when tables began to confirm it. The case of elliptic curves with complex multiplication was proved by Shimura in 1964. This conjecture stood for decades before being proved in generality.
In fact the Langlands program (or philosophy) is much more like a web of unifying conjectures; it really does postulate that the general theory of automorphic forms is regulated by the L-groups introduced by Robert Langlands. His principle of functoriality with respect to the L-group has a very large explanatory value with respect to known types of lifting of automorphic forms (now more broadly studied as automorphic representations). While this theory is in one sense closely linked with the Taniyama–Shimura conjecture, it should be understood that the conjecture actually operates in the opposite direction. It requires the existence of an automorphic form, starting with an object that (very abstractly) lies in a category of motives.
Another significant related point is that the Langlands approach stands apart from the whole development triggered by monstrous moonshine (connections between elliptic modular functions as Fourier series, and the group representations of the Monster group and other sporadic groups). The Langlands philosophy neither foreshadowed nor was able to include this line of research.
Read more about this topic: Unifying Theories In Mathematics
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