The two major trends in number theory, automorphic and arithmetic, were initiated hand in hand in the mid nineteenth century by Kronecker and Kummer. One of Kronecker’s legacies is the theory of arithmetic elliptic modular functions (and modular forms) having well-determined algebraic values (up to a specific transcendental factor: “period”) at special points on modular curves. Shimura varieties (including modular curves) were principally studied by Shimura and Deligne in the later part of the last century. In particular, the theory gives a foundation of the rationality of automorphic L-values, because Shimura varieties often supply enough rationality for us to be able to pinpoint the transcendental factors of specific L-values (the automorphic periods).
KeywordsModular Form Abelian Variety Automorphic Form Galois Representation Newton Polygon
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