Zeros of 2-adic L-functions and congruences for class numbers and fundamental units

We study the imaginary quadratic fields such that the Iwasawa λ2-invariant equals 1, obtaining information on zeros of 2-adic L-functions and relating this to congruences for fundamental units and class numbers. This paper explores the interplay between zeros of 2-adic L-functions and congruences for fundamental units and class numbers of quadratic fields. An underlying motivation was to study the distribution of zeros of 2-adic L-functions, the basic philosophy being that the location of the zeros causes restrictions on the 2-adic behavior of the class numbers and fundamental units of real quadratic fields. Though the predicted restrictions involved the unit and class number together, numerical computations (we used PARI) revealed definite patterns for the unit and class number separately, which we were then able to prove. Several of these congruences are classical, but some of them seem to be new. We use the information obtained to study the distribution of the zeros, in particular their distances from 1 and 0. In a previous paper [14], one of us showed that, if (2 + 1)/3 is prime infinitely often, then it is possible to have zeros of 2-adic Lfunctions arbitrarily close to s = 1. Recently, Morain [7] showed that (2 +1)/3 is prime, which yields a 2-adic L-function with a zero β satisfying |β−1|2 = 2−6194 (see the discussion following Theorem 5). In previous papers [12], [15], one of the present authors studied zeros of 3-adic L-functions in a somewhat similar approach. However, the advantage of using 2adic L-functions for quadratic fields Q( √ m) is that not only is the number of zeros bounded by λ−, the Iwasawa invariant for the cyclotomic Z2-extension of Q( √−m), but also there is a simple formula for λ− due to Y. Kida [6] and B. Ferrero [4]. This allows us to keep the number of zeros under control. In fact, throughout the present paper we restrict ourselves to the case λ− = 1, so we are dealing with at most one zero. 1. 2-adic L-functions Let χ be the non-trivial Dirichlet character associated to the real quadratic field Q( √ m), where m is taken to be squarefree. The 2-adic L-function L2(s, χ) satisfies Received by the editor October 14, 1997. 1991 Mathematics Subject Classification. Primary 11R11; Secondary 11S40.