The Type II Diabetes Prevention Pyramids. International Diabetes Center, 1995. From International Diabetes Center, 3800 Park Nicollet Blvd., Minneapolis, MN 55416-2699, (612) 993-3393, 21 pp, booklet, $3.95.

Abstract

In this paper, following Perrin-Riou's work, for any eigenform f of weight at least 2 and p-adic slope less than 1, we construct algebraic p-adic L-function Lf(X), and show that for eigenforms f2 of weight 2 and fk of weight k, if f2≡fk(modpN′) and ap(f2)/p≡ap(fk)/p(modpN⋅p) for certain N,N′, then for every Dirichlet character χ of sufficiently large p-power conductor, χ(Lf2(X)) and χ(Lfk(X)) have the same p-adic valuation. Combined with our previous work with Choi on analytic congruences ([2]), this implies the following: Suppose E is an abelian variety over Q associated to f2, ap(E)=0 (always true if E is an elliptic curve over Q and has good supersingular reduction at p>3), and the above conditions for f2 and fk hold true with sufficiently large N,N′ (where the meaning of “sufficiently large” depends only on E). Then, the Main Conjecture of Iwasawa Theory for fk implies the Main Conjecture for E.