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- The Fourier expansion of an elliptic modular form has been fruitfully utilized in various arithmetical problems as well as in the study of the analytic properties of the form itself. The same can be said also for the Hilbert and Siegel modular forms.
- (1978). "The arithmetic of forms with respect to a unitary group". Annals of Mathematics 107: 569–605.
Quotes about ShimuraEdit
- The subject that might be called “explicit class field theory” begins with Kronecker’s Theorem: every abelian extension of the field of rational numbers Q is a subfield of a cyclotomic field Q(ζn), where ζn is a primitive nth root of 1. In other words, we get all abelian extensions of Q by adjoining all “special values” of e(x) = exp(2πix), i.e., with x ∈ Q. Hilbert’s twelfth problem, also called Kronecker’s Jugendtraum, is to do something similar for any number field K, i.e., to generate all abelian extensions of K by adjoining special values of suitable special functions. ... Aside from some results of Hecke in 1912, the only progress on the twelfth problem was made by Shimura and Taniyama in the 1950s. They achieved complete results concerning the abelian extensions of number fields arising from abelian varieties, with complex multiplication, of arbitrary dimension n ...
- Andrew Ogg: (1999). "Review of Abelian varieties with complex multiplication and modular functions by Goro Shimura". Bull. Amer. Math. Soc. (N.S.) 36: 405–408. DOI:10.1090/S0273-0979-99-00784-3.