%0 Journal Article
%T Using Continued Fractions to Compute Iwasawa Lambda Invariants of Imaginary Quadratic Number Fields
%A Jordan Schettler
%J Journal of Numbers
%D 2014
%R 10.1155/2014/803649
%X Let be a prime such that and has class number 1. Then Hirzebruch and Zagier noticed that the class number of can be expressed as where the are partial quotients in the ¡°minus¡± continued fraction expansion . For an odd prime , we prove an analogous formula using these which computes the sum of Iwasawa lambda invariants of and . In the case that is inert in , the formula pleasantly simplifies under some additional technical assumptions. 1. Notation and Assumptions Let be a real quadratic number field of discriminant . Suppose where , are the discriminants of quadratic number fields , , respectively. We will frequently make the following assumption. Assumption A. Suppose the class number of is 1 and that is divisible by a prime congruent to modulo . Remark 1. Make Assumption A. Then has no units of negative norm and the factorization in (1) is unique (up to ordering of factors) with , negative by classical genus theory. Without loss of generality, is a prime congruent to modulo and is either , , or a prime congruent to modulo . For let denote the class number of . For a prime , let denote the Iwasawa lambda invariant of the cyclotomic -extension of . Goal. Under Assumption A, we want a formula for the sum of lambda invariants which is analogous to Hirzebruch and Zagier¡¯s formula for the product of class numbers given in terms of the partial quotients in the ¡°minus¡± continued fraction expansion of where with . To accomplish this goal, we first recall some computations of special values of partial zeta functions obtained by Kronecker limit formulas at or by the methods of Takuro Shintani at . Then we relate these to special values of -functions which can be alternatively given in terms of the arithmetic invariants and . 2. Special Values of Partial Zeta Functions Suppose is a modulus of where we view as an ideal in the ring of integers and we will always assume is the product of both real places of . We denote the narrow ray class group associated with by as in [1]. Consider the partial zeta function associated with some , that is, the meromorphic continuation of the sum where denotes the absolute norm of . We have a Laurent expansion where is a constant which depends on but not on . Computations of are called ¡°Kronecker limit formulas¡± for real quadratic number fields because Leopold Kronecker first computed this quantity in the context of an imaginary quadratic number field. If is a nontrivial character on , the -function has special values at given by We will state some results which express , in terms of continued fractions, and, in order to do so, we
%U http://www.hindawi.com/journals/jn/2014/803649/