Abstract
Let l>3 be a prime such that l≡3 (mod 4) and Q(l) has class number 1. Then Hirzebruch and Zagier noticed that the class number of Q(-l) can be expressed as h(-l)=(1/3)(b1+b2+⋯+bm)-m where the bi are partial quotients in the “minus” continued fraction expansion l=[[b0;b1,b2,…,bm¯]]. For an odd prime p≠l, we prove an analogous formula using these bi which computes the sum of Iwasawa lambda invariants λp(-l)+λp(-4) of Q(-l) and Q(-1). In the case that p is inert in Q(-l), the formula pleasantly simplifies under some additional technical assumptions.
Highlights
Where D1, D2 are the discriminants of quadratic number fields K1, K2, respectively
Suppose the class number of K is 1 and that D is divisible by a prime congruent to 3 modulo 4
Under Assumption A, we want a formula for the sum of lambda invariants λp(D1) + λp(D2) which is analogous to Hirzebruch and Zagier’s formula for the product of class numbers h(D1)h(D2) given in terms of the partial quotients in the “minus” continued fraction expansion of (δ + √D)/2 where δ ∈ {0, 1} with D ≡ δ
Summary
Where D1, D2 are the discriminants of quadratic number fields K1, K2, respectively. We will frequently make the following assumption. Under Assumption A, we want a formula for the sum of lambda invariants λp(D1) + λp(D2) which is analogous to Hirzebruch and Zagier’s formula for the product of class numbers h(D1)h(D2) given in terms of the partial quotients in the “minus” continued fraction expansion of (δ + √D)/2 where δ ∈ {0, 1} with D ≡ δ (mod 4). To accomplish this goal, we first recall some computations of special values of partial zeta functions obtained by Kronecker limit formulas at s = 1 or by the methods of Takuro Shintani at s = 0. We relate these to special values of L-functions which can be alternatively given in terms of the arithmetic invariants h(Di) and λp(Di)
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