Using Continued Fractions to Compute Iwasawa Lambda Invariants of Imaginary Quadratic Number Fields

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