Abstract of the talk given at Luminy, June 18th, 2002:

CM Periods and the anticyclotomic main conjecture

Haruzo Hida^{1}

Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A.

The two major trends in Number theory, automorphic and arithmetic, were initiated hand in hand in the mid 19th century by Kronecker and Kummer. One of Kronecker’s legacies now is the theory of automorphic periods principally studied by Shimura and Deligne in the later part of the last century. The periods not only give the canonical transcendental factor of the automorphic L–value (hereby defining its algebraic part) but also have intricate algebraic relations coming from a deep geometric heuristic. These period relations modulo algebraic numbers have been conjectured by G. Shimura and basically proven by H. Yoshida after the endeavor of many outstanding mathematicians. In the lecture, I proposed an integral version of the period conjecture (describing the relations modulo algebraic p–integers).

On the other hand, the arithmetic side of Number theory heavily influenced by Kummer’s work is represented by Iwasawa’s theory which interprets the algebraic part of the L–value as the size of a naturally given group defined by purely algebraic means. The origin of this thought goes back to the class number formula of Dirichlet, Kummer and Dedekind, which gives the size of the class group of a number field as a special value of an abelian L–function (up to a power of the period 2πi of the exponential function). The main conjecture in Iwasawa’s theory predicts precisely the same effect, replacing the class group by the so-called “Selmer group” and its “Iwasawa module” attached to arithmetic Galois representations and abelian L–functions by automorphic p–adic L–functions.

Though these two trends looked rather apart recently, the striking idea of Wiles (which has proven two of the most fundamental conjectures in Number theory) has spanned a bridge between the two thoughts. By using results of K. Fujiwara generalizing the work of Taylor-Wiles to the Hilbert modular case, we describe a close relation between an integral version of Shimura’s period conjecture (applied to CM forms) and the anticyclotomic main conjecture for CM fields. If we could prove the integral CM period relation, the main conjecture follows in most of the cases. Conversely the main conjecture implies an essential part of the integral period relations. We hope to prove the main conjecture by Iwasawa theoretic methods combined with automorphic techniques, and as a consequence, we hope to eventually come up to the proof of the period conjecture in foreseeable future.

1partially supported by the NSF grant: DMS 9988043