In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa (1959) (岩澤 健吉), as part of the theory of cyclotomic fields.
Iwasawa appeared as a minor character in the Angel Beats anime, disappearing at the end of the third episode. Having come to terms with her past, she no longer had any need to remain in the world and Yui Yoshioka eventually replaced her as the lead guitarist and singer.
One of Iwasawa’s main theorems shows that there ≥ is some degree of regularity in the behavior of h(p) Fn. We will discuss this and other theorems of Iwasawa concerning ClFn[p∞] in chapter 2.
The connection between Iwasawa polynomials and Rie-mann zeta functions can be made rigorously and is the content of the Iwasawa main conjecture. This conjecture is much deeper and was first proved by Mazur-Wiles in 1984 and reproved by Rubin in 1994 using Euler system.
Iwasawa theory began in the late 1950s with a series of investigations by Ken-kichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to p-adic L-functions.
The link between the l-adic zeta functions and the Iwasawa modules is given by the so-called “main conjecture” in Iwasawa theory, in many cases a theorem with strikingly far reaching arithmetic applications.
In this paper we give an overview of some aspects of Iwasawa theory for modular forms. We start with the classical formulation in terms of p p -adic L L -functions in the ordinary case and the ± ± -formulation for supersingular elliptic curves.
It’s worth understanding how Iwasawa’s argument goes, as through it one obtains a form of the main conjecture free from L-functions asserting an equal-ity of characteristic ideals of Iwasawa modules com-ing from unit and class groups.
Kenkichi Iwasawa was a Japanese mathematician who worked in algebraic number theory.
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