Journal of Numbers
Volume 2014 (2014), Article ID 803649, 10 pages
http://dx.doi.org/10.1155/2014/803649

Research Article

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

Jordan Schettler

Department of Mathematics, University of California, South Hall, Room 6607, Santa Barbara, CA 93106-3080, USA

Received 18 August 2014; Accepted 27 October 2014; Published 17 November 2014

Academic Editor: Cheon S. Ryoo

Copyright © 2014 Jordan Schettler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 need a couple of lemmas which we do not prove here. See [2, 3].

Lemma 2. Suppose . There is a unique “minus” continued fraction expansion (this “minus” expansion is related to the usual “plus” continued fraction expansion where and : as sequences = )

where and . Moreover, this expansion is eventually periodic if and only if is algebraic of degree . In particular, for with ,

where the bar signifies the repeating block of minimal length ; moreover, and we have a palindrome

Lemma 3. Let . For each integral ideal , there are totally positive such that

where with = Galois conjugate of . This condition on ensures that its “minus” continued fraction is purely periodic of some minimal period :

Moreover, the sequence is determined up to cyclic permutation by .

2.1. Meyer’s Theorem

Meyer studied the case in [4]. He expressed as an integral involving a logarithm of the Dedekind -function using methods earlier applied by Erich Hecke to wide ideal classes. The advantage of considering narrow ideal classes is that if there are no units of negative norm, every wide ideal class is the disjoint union of two narrow ideal classes , and the transformation properties of the Dedekind- function can then be used to explicitly evaluate the difference as multiplied times an expression involving a Dedekind sum. Friedrich Hirzebruch and Don Zagier noticed that this expression could be written as a sum of partial quotients in a certain “minus” continued fraction.

Definition 4. We denote by the set of such that for all prime ideals of dividing . Define where is any positive element of whose Galois conjugate is negative. For each , we take .

Theorem 5 (Meyer, as restated by Hirzebruch and Zagier). Suppose and that has no units of negative norm. In the notation of Lemma 3, we have

2.2. Yamamoto’s Theorem

Meyer’s theorem is sufficient to derive the known formula for class numbers, and there are generalizations which compute , for an arbitrary . For example, Yamamoto proved such a Kronecker limit formula for narrow ray classes in [5]; he further computed , using the methods of Shintani in [6]. We will find it more convenient to use these computations at because they are rational numbers and have considerably simpler descriptions in the general case.

Definition 6. Let . Choose , , as in Lemma 3. There are unique rational numbers such that

so if we extend by periodicity, we may recursively define for all integers by

where denotes the fractional part of a real number .

Definition 7. Let denote the totally positive units in with generator . Take to be the unique generator of which is greater than , so for some nonnegative integer .

Theorem 8 (Yamamoto). Let . Then

where , are Bernoulli polynomials and , , , are as in Definitions 6, 7.

3. The Formula for Class Numbers

In this section, we use the results stated above to outline a proof of the formula for class numbers due first in a special case to Hirzebruch (in [7]) and then in the more general form below to Zagier (in [2]). We do this in order to motivate the formula for Iwasawa lambda invariants.

We say is a genus character when is a real valued character on with . In this case, we can use either Meyer’s Theorem 5 or Yamamoto’s Theorem 8 to compute class numbers by factoring into the product of Dirichlet -functions.

Theorem 9 (Kronecker). Let denote the number of distinct prime factors of . Then there are exactly ways to factor up to order as in (1). Such factorizations are in bijection with the set of nontrivial genus characters . Under this correspondence,

where each is the quadratic character of .

Theorem 10 (Hirzebruch). Make Assumption A. Then

where as in Lemma 2 and each is the number of roots of unity in .

Proof. Take . By Remark 1 and Theorem 9, there is a unique nontrivial genus character on , so (15) and the analytic class number formula imply

or via functional equations

Here we have simply . Consider the trivial class in the context of Lemma 3: we can choose , , and

via Lemma 2. It is also clear that as in Definition 6, so for all . Thus Meyer’s Theorem 5 implies

and Yamamoto’s Theorem 8 implies

Combining either (20) with (17) or (21) with (18) will yield the desired result.

4. The Formula for Iwasawa Lambda Invariants

Fix a prime and number field . Let denote the cyclotomic -extension of (we will not consider any noncyclotomic -extensions in this paper); that is, is the unique subfield of

such that is isomorphic to the group of -adic integers where is some fixed algebraic closure and each a primitive th root of unity. The subfields of which contain all lie in a tower

where

The -parts of the class numbers of these intermediate fields become regularly behaved.

Theorem 11 (Iwasawa’s growth formula). There are integers , , such that class numbers of satisfy

for all sufficiently large where denotes the -adic order.

Here is a short list of what is known and conjectured about the Iwasawa invariants , , which appear in the growth formula.

(i)Iwasawa conjectured that for all and .(ii)Ferrero and Washington proved that for all when is abelian (see [8]).(iii)If has only one prime lying over and does not divide the class number of , then (see [9]).(iv)If splits completely in , then where is the number of complex places of (see, e.g., [10]).(v)Greenberg conjectured that for all primes when is a totally real number field (see [11]).

Suppose now that is a quadratic number field of discriminant , and write

Thus we always have , and conjecturally when . Assume now that . Then for infinitely many primes . Using the analogy between number fields and function fields over finite fields, Ralph Greenberg (after Theorem  3.3 in [10]) suggests the possibility that for a the lambda invariants could be bounded as varies over all primes; however, neither the boundedness nor unboundedness of the set is known for any single . On the other hand, it is conjectured that is unbounded for as ranges over all discriminants of imaginary quadratic number fields. Ferrero (see [12]) and Kida (see [13]) proved that for we have

where the sum ranges over all odd primes dividing . In particular, this shows that is unbounded. For odd , there seems to be no simple formula like (27) to compute . We will derive a formula for under Assumption A which is analogous to the formula (16) for class numbers. We first need to recall how the lambda invariant in the growth formula (25) is related to special values of -functions. We assume here that and is odd for simplicity. Let denote the quadratic character for . For each integer , choose a Dirichlet character of conductor and order ; equivalently, generates the th level in the cyclotomic -extension of . By a theorem of Kubota and Leopoldt, there is a -adic analytic function on the disk in such that

for all integers where is the Teichmüller character. In fact, there is an interpolating power series such that

for all where is a primitive th root of unity. Setting in the previous two equations gives

We define lambda and mu invariants of the power series

as follows:

On can use (30) to prove the growth formula (25) for (see, e.g., [14] or much earlier [15]), and, in fact,

Here we are using the assumption that is odd; we get a different computation for when . We compute

where is the Euler totient function. The inequality is an equality if the minimum is assumed by exactly one member of the set. In particular,

whenever . Note that we always have

Thus letting denote the unique prime ideal lying above in , we get

whenever . There is a partial converse to this statement which follows from the same observations; namely, if , then we must also have .

At this point, we should remark that the special values can be computed with generalized Bernoulli numbers via the formula

In particular,

where is the conductor of . This shows that is an algebraic integer by the work of Carlitz in [16]. However, we will compute this special value in a different way by using Yamamoto’s Theorem 8.

Factor as in (1), and suppose each . As above, let denote the quadratic character for . Then (37) implies that for sufficiently large (which we fix for the following discussion) we have

where

is the -function for a character on the narrow ray class group of the real quadratic number field with modulus . For a prime ideal of with and we have

where is the nontrivial genus character associated with the factorization and is the residue degree of over . Thus for a nonzero ideal in we have where is the absolute norm of , so for all . Suppose now that is divisible by a prime congruent to 3 modulo 4. Then the narrow ray class group is an internal direct product where is as in Definition 4 and is the kernel of the natural homomorphism . We have , so for all . Thus since we get

Suppose, in addition, that the class number of is 1. We claim there is an exact sequence

where the first map sends the fundamental unit to its congruence class modulo and the second map sends the congruence class modulo of a totally positive to the class of the principal ideal . (Note that every congruence class modulo has a totally positive representative, and any two such representatives for the same congruence class will generate the same narrow ray class.) It is clear that the second map in (44) onto since every class in is represented by an integral ideal prime to , but such an ideal must be principal (since has class number ) with totally positive generator (since the class of is trivial in ). Suppose is trivial in for some totally positive relatively prime to . Then for some totally positive , but we can write for some relatively prime to . Since both , are totally positive, we have for some . Hence in , but since , so is in the image of the first map. This proves exactness of the sequence in (44) as claimed.

Now consider a fixed class for some totally positive relatively prime to . In the context of Lemma 3 with , we may take and so that

with

where as in Lemma 2. Write

with . Define

so that condition (12) is satisfied. As per (13), we have

and it follows that for all

where and are the numerator and denominator, respectively, of the th convergent for with , by convention. Then Yamamoto’s Theorem 8 implies

where is the order of modulo and

does not depend on the class .

Let be a primitive root modulo all powers of , so, in particular, has order in . Let denote the order of modulo . Then where the sign is + or − when is inert or split, respectively, in . We have an isomorphism of abelian groups

Note that if for some integers , then we get a congruence of norms , so is either or modulo . Hence the subgroup has order or depending on whether there does or does not, respectively, exist an integer such that ; the existence of such a is equivalent to the statement that . We will often make the following simplifying assumption.

Assumption B. Suppose that . (Of course, the statement in the assumption does not always hold; e.g., if and , then has order modulo 7, and, in fact, .)

Making Assumption B implies is the order of modulo , and thus the quotient group is cyclic of -prime order where if and otherwise. Choose a totally positive whose congruence class generates this quotient. In fact, we may choose any whose congruence class generates this quotient since the quantities can be modified modulo . Then we have a surjection

given by which is either one-to-one or two-to-one depending on whether or , respectively. For each write

where , , and then define

for each . Note that since has prime-to- order modulo , so (43) and (51) imply that

To ease notation we define a twisted, homogeneous Dedekind sum for an arbitrary Dirichlet character of modulus :

Since the character also generates the th level in the cyclotomic -extension of and since runs through the units modulo as runs through , we have proved the following.

Theorem 12. Make Assumption A. Suppose is an odd prime satisfying Assumption B. Then for sufficiently large

where as in Lemma 2 and the rest of the notation is as above.

Remark 13. We can compute the Dedekind sums for any integer as follows. Write where and . If , then since for all . Thus we may assume . Choose with . Then

If (i.e., ), then and , so

since

On the other hand, if (i.e., ), then for a primitive root modulo all powers of as above, we have

where and is a primitive th root of unity. Thus , so (61) holds in this case as well since when . We summarize the results of this remark in the following proposition.

Proposition 14. One has for all

In light of the above, one might hope to also evaluate the sums

using a reciprocity law for Dedekind sums with characters, but the author is presently unaware of how this can be done. Nonetheless, Theorem 12 provide us with a means of computing lambda invariants.

Example 15. Take and let be a prime such that the number field of discriminant has class number one. Then the totally positive fundamental unit in has order dividing in . Suppose this order is exactly and that . Then for any positive integer we have that is a primitive root modulo and that is cyclic of order since the order of modulo will be . We want a generator of this quotient, and it clearly suffices to choose to be an element of order in

Alternatively, we may regard as an eighth root of unity in , and in that case, a fixed choice of will suffice for all . We can construct such an by using Hensel lifting on since

Let us consider now a concrete case. Take . Then

so

It is easy to check that has order modulo since

and . We also easily verify that since

For , we compute the right hand side of (59) and get

where . Of course, since 3 remains inert in , so we must have by the comments following (37). Likewise, for we get

where , so . For , we find

where . Thus . This and other similar computations (with the help of gp/pari) agree with known results as found in [17], for example.

Under additional assumptions, we can compute lambda invariants without having to compute an as above. In particular, the invariants can be computed using only a choice of primitive root and the mod data from the continued fraction expansion of .

Corollary 16. Suppose is a prime such that has class number . For each let and denote the numerator and denominator, respectively, of the th convergent in the “minus” continued fraction expansion where is minimal period and , by convention. Let be an odd prime such that is inert in and that the fundamental unit has order modulo satisfying the following technical assumptions:

(1),(2) if ,(3) if .

Choose to be a primitive root modulo all powers of , so there are integers , with and . For all take , to be the least nonnegative residues of , modulo , and let , denote the unique integers such that and . Then for sufficiently large ,

where is a primitive th root of unity.

Proof. Obviously, Assumptions  A,  B hold for and , so we may apply all of the ideas which culminated in Theorem 12. In particular, we will exhibit a set of representatives for modulo . The assumption that is inert in is equivalent to the statement that is not a square modulo . Thus for with we know that is a unit modulo which is never congruent to a power of modulo since otherwise , a contradiction. Now we use our technical assumptions on . In the case that , we have assumed that , so is inert in and there is a unique element of order two in which corresponds to and is a power of modulo . Consider the map

given by the restriction of the map in (44). This map is two-to-one in the case just described. Similarly, in the case that , we have assumed that is odd, so is split in and now the map in (76) is one-to-one since is not congruent to a power of modulo in this case. By (43), (50), and (51), we get for sufficiently large that is

where with and .
We have the formula for all integers , , so the sequences , are periodic modulo with period . Thus for all , the sequences , are periodic with period , so

The result follows.

Remark 17. Also, we note that the periodicity also implies that , are “palindromic” in the following sense:

As remarked above, the sequence for is periodic with period and is “palindromic” with for while . Hence if , then , so

and similarly . This implies that we can replace the upper index of the sum on with and still maintain the same -adic order. Of course, we could for the same reason ignore the in the denominator of our sum in the corollary, but it is natural to include this factor of since

In fact, since the map in (76) is two-to-one when , we can replace the upper index on with in this case and still conclude the sum is in with the same -adic order.

Example 18. Let be a Fermat prime and let be a prime such that is inert in and has class number . Then is quadratic nonresidue modulo , so here we can choose assuming additionally that . In this case is a primitive th root of unity with , so we do not have to worry about computing here. For fixed, these conditions on are just congruence conditions modulo plus the assumption that has class number 1, so there should be many such examples. We just need to check the conditions on the fundamental unit in these cases in order for the corollary to apply.
Let us consider the concrete case of and . Then is inert in both (class number ) and . We have

so , the class number of is , and the fundamental unit of is

It is easy to check that has order modulo and that . For , we compute

Hence . Since divides the class number of and is split in , we must have both and , so .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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