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We provide a new insight of the difficulty of nonparametric estimation of a whole function. A new method is invented for finding a minimax lower bound of globally estimating a function. The idea is to adjust automatically the direction to the nearly hardest I-dimensional subproblem at each location, and to use locally the difficulty of I-dimensional subproblem. In a variety of contexts, our method can give not only attainable global rates, but also constant factors. Comparing with the existing techniques, our method has the advantages of being easily implemented and understood, and can give constant factors as well.

For primes q≡7 mod 16, the present manuscript shows that elementary methods enable one to prove surprisingly strong results about the Iwasawa theory of the Gross family of elliptic curves with complex multiplication by the ring of integers of the field K=Q(√(-q)), which are in perfect accord with the predictions of the conjecture of Birch and Swinnerton-Dyer. We also prove some interesting phenomena related to a classical conjecture of Greenberg, and give a new proof of an old theorem of Hasse.